Semigroup decay for the wave equation with unbounded damping

SEMIGR OUP DECA Y F OR THE W A VE EQUA TION WITH UNBOUNDED D AMPING ANTONIO ARNAL, BORBALA GERHA T, JULIEN RO YER, AND PETR SIEGL Abstract. W e study the damped w av e equation with a damping coeﬃcient which is p ossibly singular and un bounded at inﬁnity . In general, zero belongs to the sp ectrum of the corresponding generator, which prev ents a uniform (exponential) deca y for the energy . Ho w ever, for initial conditions in a suitable subspace, a detailed analysis of the resolv ent norm for low frequencies leads to sharp polynomial time-decay rates for the solution and its energy . 1. Introduction and main resul t Consider the damp ed wa ve equation (D WE)      ∂ tt u ( t, x ) + a ( x ) ∂ t u ( t, x ) = (∆ − q ( x )) u ( t, x ) , t > 0 , x ∈ Ω , u ( t, x ) = 0 , t > 0 , x ∈ ∂ Ω , ( u (0) , ∂ t u (0)) = ( f , g ) , (1.1) where ∅  = Ω ⊂ R d is open and the non-negativ e damping co eﬃcien t (absorp- tion index) a and non-negativ e p otential q satisfy a minimal regularity assumption (i.e. only lo cal integrabilit y): 0 ≤ a, q ∈ L 1 loc (Ω) . (1.2) Throughout this pap er, w e will consider the unique mild solution u ( t ) for the prob- lem (1.1) (in the sense given b y Prop osition 2.1 and Remark 2.2 below). Our goal is to obtain the decay rates of several quan tities, in particular the energy E ( u ; t ) = ∥∇ u ∥ 2 L 2 + ∥ q 1 2 ∇ u ∥ 2 L 2 + ∥ ∂ t u ∥ 2 L 2 as well as (weigh ted) L 2 -norms of the solution u ( t ), dep ending on the co eﬃcien ts a , q and the initial conditions. W e are mainly interested in the case where Ω is un b ounded and the co eﬃcien ts a and q ma y b e large at inﬁnit y . W e focus here on the case of uniformly p ositiv e a , whic h simpliﬁes the analysis of high frequencies (but is not essen tial for our k ey lo w frequency estimates; see Section 7 where further results and commen ts on the non-uniformly positive case are discussed). An example of (regular) co eﬃcien ts illustrating the setting is Ω = R d , a ( x ) = | x | 2 + 1 , q ( x ) ≡ q 0 ≥ 0 , x ∈ Ω . (1.3) Date : March 24, 2026. 2010 Mathematics Subje ct Classiﬁc ation. 35L05, 35P05, 34G10, 34L40, 47A10, 47D06, 34D05, 35B40, 26A12. Key wor ds and phrases. damp ed wa ve equation, unbounded damping, resolven t b ounds, lo w frequencies, semi-uniform stability . A. Arnal ackno wledges the support of NA WI Graz for his p ostdoctoral stay at TU Graz in 2023-2024. This research was partially funded by the Austrian Science F und (FWF) 10.55776/P 33568-N. B. Gerhat has received funding from the European Union’s Horizon 2020 research and innov ation programme under the Marie Sk lodowsk a-Curie Gran t Agreement No. 101034413, and the EXPR O gran t No. 20-17749X of the Czec h Science F oundation. J. Royer is supp orted by the Labex CIMI, T oulouse, F rance, under gran t ANR-11-LABX-0040-CIMI. W e are grateful to P erry Kleinhenz for numerous inspiring discussions. 1 2 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL Main results. In order to state our results, w e in tro duce the energy Hilb ert space H := W ⊕ L 2 (Ω) , ∥ F ∥ 2 H := ∥ f ∥ 2 W + ∥ g ∥ 2 L 2 , F = ( f , g ) ∈ H . (1.4) Here W is the Hilbert completion of C ∞ c (Ω) with resp ect to the inner pro duct ⟨· , ·⟩ W inducing the norm ∥ u ∥ 2 W := ∥∇ u ∥ 2 L 2 + ∥ q 1 2 u ∥ 2 L 2 , u ∈ W . In general, W might not b e contained in L 2 (Ω) but, in an y case, ∇ u and q 1 2 u are (iden tiﬁed with) functions in L 2 (Ω) (see App endix A for more details). The essen tial ﬁrst observ ation is that a uniform (and hence exponential) energy deca y of solutions is p ossible only in the case when the damping co eﬃcien t a is dominated by − ∆ + q in the sense of (1.6) below. Prop osition 1.1. L et Ω ⊂ R d b e non-empty and op en, and let 0 ≤ a, q ∈ L 1 loc (Ω) . If ther e exist ν > 0 and C > 0 such that for al l F = ( f , g ) ∈ H and t ≥ 0 the solution u ( t ) of (1.1) satisﬁes ∥∇ u ( t ) ∥ L 2 + ∥ q 1 2 u ( t ) ∥ L 2 + ∥ ∂ t u ( t ) ∥ L 2 ≤ C e − ν t ∥ F ∥ H , (1.5) then ther e exists M > 0 such that ∥ a 1 2 ϕ ∥ 2 L 2 + ∥ ϕ ∥ 2 L 2 ≤ M  ∥∇ ϕ ∥ 2 L 2 + ∥ q 1 2 ϕ ∥ 2 L 2  , ϕ ∈ C ∞ c (Ω) . (1.6) Mor e over, if a ( x ) ≥ a 0 > 0 a.e. in Ω , then the r everse implic ation holds, so that (1.5) and (1.6) ar e e quivalent in this c ase. Here we are in terested in cases lik e (1.3), where (1.6) do es not hold and uniform energy deca y cannot prev ail. In our main theorem below, w e hence consider initial conditions in a subspace of H , namely in the Hilb ert space K :=  ( f , g ) ∈  H 1 0 (Ω) ∩ Dom( q 1 2 )  × L 2 (Ω) : af ∈ L 1 loc (Ω) , af + g ∈ W ∗  , (1.7) endo wed with the norm ∥ F ∥ 2 K = ∥ ( f , g ) ∥ 2 K := ∥ ( f , g ) ∥ 2 H + ∥ f ∥ 2 L 2 + ∥ af + g ∥ 2 W ∗ , (1.8) (see Section 2.4 b elo w). Since for F ∈ K w e ha v e af + g ∈ L 1 loc (Ω)  → D ′ (Ω), the condition af + g ∈ W ∗ means that af + g has an extension to a b ounded functional on W (whic h is unique due to the densit y of C ∞ c (Ω) in W ). Our main result reads as follo ws. Theorem 1.2. L et Ω ⊂ R d b e non-empty and op en, let 0 ≤ a, q ∈ L 1 loc (Ω) and let a ( x ) ≥ a 0 > 0 a.e. in Ω . Then ther e exists C > 0 such that, for any initial data F ∈ K and al l t ≥ 0 , the solution u ( t ) of (1.1) de c ays in time as ∥∇ u ( t ) ∥ L 2 + ∥ q 1 2 u ( t ) ∥ L 2 ≤ C ∥ F ∥ K ⟨ t ⟩ − 1 , (1.9) ∥ ∂ t u ( t ) ∥ L 2 ≤ C ∥ F ∥ K ⟨ t ⟩ − 3 2 , (1.10) ∥ a 1 2 u ( t ) ∥ L 2 + ∥ u ( t ) ∥ L 2 ≤ C ∥ F ∥ K ⟨ t ⟩ − 1 2 . (1.11) Mor e over, if Ω is an exterior domain and, for some β > 0 and c > 0 , a ( x ) ≥ c ⟨ x ⟩ β a.e. in Ω , then, for al l t ≥ 0 , ∥ ∂ t u ( t ) ∥ L 2 ≤ C ∥ F ∥ K ⟨ t ⟩ − 3 2 − β 2(2+ β ) , (1.12) ∥ u ( t ) ∥ L 2 ≤ C ∥ F ∥ K ⟨ t ⟩ − 1 2 − β 2(2+ β ) . (1.13) SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 3 Notice that, combining (1.9) and (1.10), the energy decays as E ( u ; t ) ≤ C ∥ F ∥ 2 K ⟨ t ⟩ − 2 , t ≥ 0 . Moreo ver, w e get a similar estimate for the L 2 -norm of u ( t ) whenev er it is con trolled b y the left-hand side of (1.9) (for instance if q is uniformly p ositiv e or if Ω is such that the Poincar ´ e inequality holds, see also Remark 3.5 b elo w). W e show in Section 6 that the estimates (1.9)–(1.11) are sharp in the mo del case Ω = R d , q ( x ) ≡ 0, a ( x ) ≡ 1 and can thus not be impro ved in the general setting. Strategy. The strategy of our pro of relies on the semigroup p oin t of view and the related spectral and resolven t analysis. More precisely , the energy deca y is obtained via res olv en t estimates for the damped wa v e op erator in K (see Section 2) sp eciﬁcally for the sp ectral parameter λ ∈ i R \ { 0 } (see Section 5 and in particular Theorem 5.2 b elow). F rom this spectral p oin t of view, w e can deal separately with the contributions of high ( λ → ± i ∞ ) and lo w ( λ → 0) frequencies. The con tribution of high frequencies is not our primary concern in this paper. F or uniformly p ositive a (even with minimal regularit y), the resolven t norm of the generator is b ounded for high frequencies (see Prop osition 4.1). Th us the uniform exp onen tial energy decay would follo w if the resolven t were also b ounded at low frequencies, which is the main issue here (see b elow). In fact, our results generalize in a straigh tforward wa y if the resolven t remains b ounded for high frequencies, whic h can hold also without a uniformly p ositiv e a . Namely , this is known in d = 1 for a regular a which is un b ounded at ±∞ and p ossibly zero on a b ounded subset of R (see [5, Thm. 3.5] or Theorem 7.1 below), as well as for the example a ( x ) = | x | β , β > 0, x ∈ R d , discussed in Section 7. These conclusions are natural as they suggest that the resolven t at ± i ∞ remains bounded for unbounded damping also in higher dimensions as long as the usual Geometric Con trol Condition (GCC) is satisﬁed. W e recall that the GCC says that all the ra ys of light (or classical tra jectories), along which high frequency w av es propagate, go through the damping region { a > 0 } (see references b elo w). On the other hand, the presence of an undamp ed tra jectory in a wa v eguide (see the example in Section 7.2) yields v arious rates of resolven t growth at ± i ∞ dep ending on the b ehavior of a in the neighborho o d of this tra jectory (in line with the conclusions in [37]). Nonetheless, the singularit y of the resolven t for high frequencies in these examples is alw a ys milder than the one around zero (see b elo w) and so the resulting energy deca y rate originates in the lo w frequencies. The main analysis in this pap er concerns lo w frequencies ( λ → 0). Note that if (1.6) is not satisﬁed, then zero is in the essential sp ectrum of the generator (see Corollary 2.7 for details and Theorem 2.3 for further claims on the real essential sp ectrum). This is the eﬀect resp onsible for non-exp onen tial rates and was noticed ﬁrst in [25]. The p olynomial rates in Theorem 1.2 are obtained from resolv ent estimates for λ → 0, which are pro ved without assuming the uniform p ositivit y of a ; for in- stance Ω = R d and a ( x ) ≥ 1 for a.e. | x | ≥ 1 is co v ered (see Assumption 3.1, Theorem 3.3 and also Remark 3.5). Although the damp ed wa ve operator is highly non-self-adjoin t for unbounded damping (see Figure 2.1 and [4]), the key observ ation is that the resolven t estimates around zero in Theorem 3.3 can be reduced to a self- adjoin t sp ectral problem. This allows for classical general to ols (Neumann brack- eting [46, Chap. XI II.15] and asymptotic p erturbation theory [33, Chap. VI I I]). Moreo ver, the recent new functional-analytic understanding of the arising opera- tors in [27] enabled us to work with the minimal L 1 loc assumptions on a and q . Literature. Most existing results deal with the case where a (and q ) are b ounded. When Ω is b ounded, then (1.6) holds b y the P oincar´ e inequalit y and lo w frequencies 4 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL are not an issue. It is prov ed in [45] (see also [44]) that the GCC is essentially necessary and suﬃcient for the uniform (exp onen tial) energy decay (see also [7] for damping at the b oundary). On the other hand, when the GCC is violated, the deca y cannot b e uniform and some regularity is required for the initial condition. Under minimal assumptions on the damping the decay is at least logarithmic in time (see [36]), and for in termediate situations, depending on the geometry of the undamp ed rays of ligh t, v arious p olynomial deca y rates hav e b een obtained (see for instance [15, 3, 37, 34, 35] and references therein). These pap ers also raise the question of the link b etw een the regularit y of the damping where it v anishes and the decay rate of the wa ve. See also [14] for a reﬁned version of the GCC for rough dampings. If q ( x ) ≡ 1 (then we refer to (1.1) as the Klein-Gordon equation) and a is b ounded, then (1.6) alw ays holds, even if Ω is unbounded, and the results are similar to the b ounded case ab ov e. See [16] for the minimal logarithmic deca y with loss of regularity under some weak assumption on a and the exp onen tial uniform deca y under a suitable v ersion of the GCC. When q ( x ) ≡ 0 and Ω is un b ounded and suc h that the Poincar ´ e inequalit y do es not hold, then (1.6) fails and the decay cannot be uniform because of the lack of deca y in the c on tribution of low frequencies. Of course, depending on the geometry and the damping, we can still hav e independently a lack of deca y in the con tribution of high frequencies. In particular, when a ( x ) ≡ 0, there is no decay at all. Nonetheless, one can consider a very closely related problem of the local energy decay (see [13, 11] without damping, [24] when the damping is small at inﬁnity , and references therein). As already mentioned, the simplest model for a damped w a ve equation without (1.6) is the case with Ω = R d and a ( x ) ≡ 1, which will b e discussed in Section 6. Estimates for the solution u ( t ) and its deriv atives in Lebesgue spaces hav e ﬁrst been pro ved in [39]. Similar results, as w ell as the diﬀusiv e phenomenon (the damp ed w av e b eha ves for large times like a solution of a heat equation, see Section 6) ha v e then b een prov ed in v arious settings. See for instance [29, 2] in an exterior domain, [53, 31, 54, 51] for slowly deca ying dampings, [48, 38] in a wa veguide and [32] in a p eriodic setting. W e also refer to [17, 42, 43, 8, 40] for results in abstract settings. In this pap er, we are interested in the wa v e equation with un b ounded dampings. W e refer to [25, 4] for earlier studied sp ectral prop erties of the w a v e op erator. Closer to our problem, Ik ehata and T akeda ha ve prov ed in [30] energy deca y as in Theorem 1.2 in the case Ω = R d with d ≥ 3, q ( x ) ≡ 0 and uniformly positive a ∈ C ( R d ). More precisely , b y emplo ying a mo diﬁed Mora w etz (multiplier) metho d, they established that for initial data satisfying F = ( f , g ) ∈  H 1 ( R d ) ∩ L 1 ( R d )  ×  L 2 ( R d ) ∩ L 1 ( R d )  , af ∈ L 1 ( R d ) ∩ L 2 ( R d ) , w eak solutions of (1.1) deca y b oth in energy and L 2 -norm as E ( u ; t ) = ∥ ∂ t u ( t ) ∥ 2 L 2 + ∥∇ u ( t ) ∥ 2 L 2 ≤ C ( F ) 2 ⟨ t ⟩ − 2 , ∥ u ( t ) ∥ L 2 ≤ C ( F ) ⟨ t ⟩ − 1 2 , t ≥ 0 , (1.14) (see [30, Thm. 1.2]). The used metho d does not apply in dimensions d = 1 , 2 and the decay in this case remained op en (see [30, Sec. 3] for a detailed explanation). Our result co v ers all dimensions d , thus it solves the op en problem, and it also allo ws for a minimal regularity of a and q as w ell as a general op en Ω instead of R d . Moreo ver, in this generalit y , the same rates as in (1.14) are obtained for the L 2 -norm of u ( t ) and its gradient, and w e ha v e a better deca y for ∂ t u ( t ). A detailed comparison of Theorem 1.2 and the result in [30], indicating also the origin of the SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 5 restriction d ≥ 3 coming from the Sobolev inequalit y , as w ell as the relation of C ( F ) and ∥ F ∥ K , can b e found in Section 8 below. In [52], Soba jima and W ak asugi obtained an impro ved estimate when the damp- ing is indeed unbounded at inﬁnity . On exterior domains Ω ⊂ R d with d ≥ 2 and ∂ Ω smo oth, q ( x ) ≡ 0 and the damping a ∈ C 2 (Ω) satisfying a > 0 on Ω and lim | x |→∞ a ( x ) | x | β = a 0 > 0 with β > 0, they prov e the following weigh ted L 2 -norm decay for a compactly supp orted initial condition F ∈ ( H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω) (with an arbitrary δ > 0 when d > 2 and δ = 0 in case that d = 2) ∥ a 1 2 u ( t ) ∥ L 2 ≤ e C ( F ) ⟨ t ⟩ − d 2(2+ β ) − β 2(2+ β ) + δ , t > 0 , (1.15) (see [52, Cor. 1.2]). Their metho d relies on the comparison with the solutions of the asso ciated heat equation and leads to a decay rate depending on the spatial dimension d . In comparison, our estimates (1.12)–(1.13) seem to b e the ﬁrst results exhibiting the eﬀect of the unboundedness of a at inﬁnity without restricting to initial data with compact supp ort (unlike (1.15), our rates are d -indep enden t). Plan of the pap er. In Section 2, w e recall ho w the generator of the damp ed wa v e equation is deﬁned in the presence of unbounded co eﬃcients, and we giv e its basic prop erties in the spaces H and K . In Section 3, we pro ve the resolv en t estimates for lo w frequencies and the asso ciated co ercivity for the Sch ur complement. Section 4 is concerned with the high frequency resolv en t estimates for uniformly positive damping. In Section 5, we sho w how we can deduce the time decay estimates from the resolv ent estimates. In Section 6, w e pro ve that our rates are sharp in the model case with constan t coeﬃcients. The optimality in this case is based on the diﬀusive phenomenon and decay estimates for the heat equation. Section 7 is devoted to the case of non-uniformly p ositiv e damping; w e discuss some cases where GCC holds and the high frequency estimates remain v alid. W e also presen t an example where the GCC is violated (but nevertheless the app earing singularity of the resolv en t do es not sp oil the deca y rate of the semigroup). Finally , w e compare in Section 8 our approac h with the previous result by Ikehata and T akeda which motiv ated our study , and some tec hnical details ab out the space W are collected App endix A at the end of the pap er. Notation. W e write C + = { λ ∈ C : Re λ > 0 } , C − = − C + , D = { λ ∈ C : | λ | < 1 } , D + = D ∩ C + . The resolven t set, sp ectrum, p oint sp ectrum and contin uous sp ectrum of a linear op erator T are denoted by ρ ( T ), σ ( T ), σ p ( T ) and σ c ( T ), re- sp ectiv ely . The essential spectra σ e j ( T ) of T , j = 1 , . . . , 5, are deﬁned as in [22, Chap. IX]. T o av oid introducing m ultiple constants whose exact v alues are inessen- tial for our purposes, we write a ≲ b to indicate that, giv en a, b ≥ 0, there exists a constan t C > 0, indep enden t of any relev ant v ariable or parameter, suc h that a ≤ C b . The relation a ≳ b is deﬁned analogously whereas a ≈ b means that a ≲ b and a ≳ b . Finally , we write ⟨ t ⟩ = √ 1 + t 2 for t ∈ R . 2. Genera tor of the d amped w a ve equa tion for unbounded damping 2.1. Deﬁnition of the generator and solution of the damped wa ve equa- tion. As describ ed in the in troduction, we study the time dependent problem (1.1) from a sp ectral p oin t of view. F ormally , w e can rewrite (1.1) as ( ∂ t U ( t ) = A U, U (0) = F , (2.1) 6 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL where U ( t ) = ( u ( t ) , ∂ t u ( t )), F = ( f , g ) and A =  0 I ∆ − q − a  . (2.2) It is natural to see A as an operator in the space H = W ⊕ L 2 (Ω) deﬁned as in (1.4), so that ∥ U ( t ) ∥ 2 H is then precisely the energy of u ( t ). Note that if Ω and q are such that ∥∇ u ∥ 2 L 2 + ∥ q 1 2 u ∥ 2 L 2 ≳ ∥ u ∥ 2 L 2 , u ∈ C ∞ c (Ω) , (2.3) e.g. when the P oincar ´ e inequalit y applies or when q ( x ) ≥ q 0 > 0 a.e. in Ω, then W is a subspace of L 2 (Ω) and coincides (with equiv alen t norms) with H 1 0 (Ω) ∩ Dom( q 1 2 ) equipp ed with the natural norm ( ∥∇ u ∥ 2 L 2 + ∥ q 1 2 u ∥ 2 L 2 + ∥ u ∥ 2 L 2 ) 1 2 , (2.4) i.e. the form domain of the self-adjoint Diric hlet realization of − ∆ + q in L 2 (Ω). Due to the minimal assumptions on the coeﬃcients (1.2), where no relativ e b oundedness (in any sense) of a with resp ect to − ∆ + q is a v ailable, the recent metho d of dominant Sch ur complemen ts [27] is employ ed to ﬁnd a realization of A whic h generates a C 0 -con traction semigroup. T o this end, we introduce the Hilb ert space D t := H 1 0 (Ω) ∩ Dom( q 1 2 ) ∩ Dom( a 1 2 ) , (2.5) equipp ed with the inner pro duct arising from ∥ u ∥ 2 D t := ∥∇ u ∥ 2 L 2 + ∥ q 1 2 u ∥ 2 L 2 + ∥ a 1 2 u ∥ 2 L 2 + ∥ u ∥ 2 L 2 . (2.6) W e deﬁne A as an op erator in H with the domain Dom( A ) =  F = ( f , g ) ∈ W × D t : (∆ − q ) f − ag ∈ L 2 (Ω)  ; (2.7) (see App endix A for details ab out W and the meaning of ( − ∆ + q ) f when f ∈ W ). Prop osition 2.1 ([27, Thm. 4.2]) . The op er ator A deﬁne d by (2.2) and (2.7) is densely deﬁne d and m-dissip ative in H , henc e it gener ates a str ongly c ontinuous c ontr action semigr oup e t A on H . Remark 2.2. Recall that for F ∈ H , the function U ( t ) = e t A F is the unique mild solution of the Cauc hy problem (2.1) in H (and note that U ( t ) is even the unique classical solution of (2.1) in H if F ∈ Dom( A )). In detail, being a mild solution in H means that U : [0 , ∞ ) → H is contin uous, its primitive satisﬁes Z t 0 U ( s ) d s ∈ Dom( A ) , t ≥ 0 , and it solves the in tegrated Cauch y problem U ( t ) = A  Z t 0 U ( s ) d s  + F , t ≥ 0 . (2.8) The ﬁrst comp onen t of the vector U ( t ) = ( u ( t ) , v ( t )) is our solution for the DWE (1.1). Considering the action of A in (2.2), it reads u ( t ) = Z t 0 v ( s ) d s + f , t ≥ 0 . This in particular implies u ∈ C 1 ([0 , ∞ ); L 2 (Ω)) , U ( t ) = ( u ( t ) , ∂ t u ( t )) , t ≥ 0 . (2.9) In Section 8, we discuss further prop erties of u ( t ) for sp ecial initial conditions F ∈ K , as w ell as its relation to the weak solutions used in [30]. SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 7 2.2. The Sc h ur complement. The space D t in tro duced in (2.5) and (2.6) arises as the domain of the forms t λ [ u ] := ∥∇ u ∥ 2 L 2 + ∥ q 1 2 u ∥ 2 L 2 + λ ∥ a 1 2 u ∥ 2 L 2 + λ 2 ∥ u ∥ 2 L 2 , Dom( t λ ) := D t , (2.10) whic h are co erciv e on D t (after a shift and rotation) for λ ∈ C \ ( −∞ , 0] (see [27, Lem. 4.13]). Thus the t λ deﬁne Schr¨ odinger op erators T λ = − ∆ + q + λa + λ 2 , Dom( T λ ) =  u ∈ D t : ( − ∆ + q + λa ) u ∈ L 2 (Ω)  , λ ∈ C \ ( −∞ , 0] , (2.11) whic h are after a shift and rotation m-sectorial. Notice that T λ is essen tially the Sc hur complement of A . W e will deduce the sp ectral prop erties of A from the analysis of T λ . W e recall the following sp ectral properties for A and T λ . Theorem 2.3 ([27, Thm. 4.2], [25, Sec. 2–4]) . L et A and T λ b e as in (2.2) , (2.7) and (2.11) , r esp e ctively. Then the fol lowing claims hold. (i) We have the sp e ctr al e quivalenc e for ∗ ∈ { , p , e2 } ∀ λ ∈ C \ ( −∞ , 0] : λ ∈ σ ∗ ( A ) ⇐ ⇒ 0 ∈ σ ∗ ( T λ ) . (2.12) (ii) If λ ∈ σ ( A ) \ ( −∞ , 0] , then λ ∈ σ ( A ) , Re λ ≤ − 1 2 ess inf x ∈ Ω a ( x ) and | λ | 2 ≥ inf ( σ (( − ∆ + q ) D )) , wher e ( − ∆ + q ) D is the self-adjoint Dirichlet r e alization of − ∆ + q in L 2 (Ω) . (iii) If T λ has c omp act r esolvent for some λ ∈ C \ ( −∞ , 0] , then σ ( A ) \ ( −∞ , 0] is pur ely discr ete, i.e. it c onsists of isolate d eigenvalues with ﬁnite algebr aic multi- plicities. In p articular, this holds if Ω is b ounde d or if lim R →∞ ess inf | x | >R, x ∈ Ω a ( x ) = + ∞ . (2.13) (iv) Supp ose in addition that a, q ∈ C 1 (Ω) and Ω c ontains a se ctor S δ :=  ( x 1 , x ′ ) ∈ R × R d − 1 : x 1 > 0 , | x ′ | < δ x 1  ⊂ Ω for some δ > 0 , and that a c an b e de c omp ose d as a ( x ) = a r ( | x | ) + a p ( x ) such that lim r →∞ a r ( r ) = ∞ , lim r →∞ a ′ r ( r ) a r ( r ) = 0 , lim x →∞ , x ∈ S δ a 2 p ( x ) + q ( x ) a r ( | x | ) = 0 . Then ( −∞ , 0] ⊂ σ e 2 ( A ) . (2.14) As an illustration of the generic sp ectral (and pseudospectral) properties for un b ounded damping which is not con trolled b y − ∆ + q , we recall an example with Ω = R , a ( x ) = 2 x 2 and q = 0 from [25, 4, 5]. Here σ ( A ) = ( −∞ , 0] ˙ ∪ n 2 1 3 e ± i 2 3 π (2 k + 1) 2 3 o k ∈ N 0 (2.15) is illustrated in Figure 2.1; further examples with explicit eigen v alues can b e found in [25, Sec. 6] and other suﬃcien t conditions on a and q so that (2.14) holds in [25, Sec. 4]. Note that when the damping is un b ounded at inﬁnit y , the resolv ent b ehavior of the damped wa v e operator in the left complex half-plane is quite non-trivial (see Figure 2.1 for illustration), and qualitativ ely resem bles Sc hr¨ odinger op erators with un b ounded complex potentials (see e.g. [19]). Nev ertheless, unlike for Sc hr¨ odinger op erators, where the resolven t behavior at ± i ∞ dep ends on the gro wth rate of the p oten tial at inﬁnit y (see e.g. [12, 41, 21, 28, 6]), the resolv en t norm of the damped w av e op erator is bounded (and in general non-decaying) at ± i ∞ , irrespective of the gro wth rate of a (see [5] and Section 7.1). 8 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL - 8 - 6 - 4 - 2 0 - 10 - 5 0 5 10 - 12 - 10 - 8 - 6 - 4 - 2 0 Figure 2.1. Figure repro duced from [4]. Numerical computation of sp ectrum (in blac k, see (2.15)) and pseudospectra (log 10 scale, approxi- mation b y 800 × 800 matrix) of A (see Section 2) with a ( x ) = 2 x 2 and q ( x ) ≡ 0, x ∈ R . 2.3. The resolven t of A . W e recall some steps in [27, Sec. 4] and in particular a represen tation of the resolven t of A in terms of its Sch ur complemen t. F or λ ∈ C \ ( −∞ , 0] the form t λ deﬁnes a b ounded distribution-v alued op erator b T λ ∈ B ( D t , D ∗ t ) , b T λ u := t λ [ u, · ] ∈ D ∗ t , u ∈ D t . (2.16) The operator T λ in (2.11) is in fact the restriction of b T λ to the maximal domain in L 2 (Ω), namely , T λ := b T λ | Dom( T λ ) . Due to the co ercivit y of t λ (without an y shift or rotation for Re λ > 0, see [27, Lem. 4.13]), w e hav e b T − 1 λ ∈ B ( D ∗ t , D t ) , Re λ > 0 . (2.17) Moreo ver, it is helpful to observe that the following b ounded extension prop erty holds (for further generalizations see [27]). Lemma 2.4. L et λ ∈ C \ ( −∞ , 0] and let T λ and b T λ b e as in (2.11) and (2.16) , r esp e ctively. If T − 1 λ ∈ B ( L 2 (Ω)) , then b T − 1 λ ∈ B ( D ∗ t , D t ) . Pr o of. Indeed, it is pro v en in [27, Lem. 4.13] that there exists a shift µ λ ∈ C such that ( b T λ − µ λ ) − 1 ∈ B ( D ∗ t , D t ). Then ( T λ − µ λ ) − 1 ∈ B ( L 2 (Ω)) and, using the ﬁrst resolv ent iden tity , we infer that T − 1 λ = ( T λ − µ λ ) − 1 − µ λ T − 1 λ ( T λ − µ λ ) − 1 ⊂ ( b T λ − µ λ ) − 1 − µ λ T − 1 λ ( b T λ − µ λ ) − 1 =: R λ ∈ B ( D ∗ t , D t ) , where for the b oundedness w e hav e used that D t ⊂ L 2 (Ω) is b oundedly embedded and T − 1 λ ∈ B ( L 2 (Ω) , D t ) (see [27, Lem. 2.12]). W e then ha ve R λ b T λ | Dom( T λ ) = I Dom( T λ ) , b T λ R λ | L 2 (Ω) = I L 2 (Ω) . By contin uity and densit y (see [27, Lem. 4.13] for the densit y of Dom( T λ ) in D t ), these identities extend from Dom( T λ ) and L 2 (Ω) to D t and D ∗ t , resp ectiv ely . This implies b T − 1 λ = R λ . □ The op erator A in H is implemented as a restriction of a b ounded distribution- v alued op erator as well. T o this end, in the second component of the pro duct space SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 9 H , we consider the triple (see (2.5) and (2.6)) D t ⊂ L 2 (Ω) b = L 2 (Ω) ∗ ⊂ D ∗ t and the distribution-v alued operator matrix is introduced as b A :=  0 I ∆ − q − a  ∈ B ( W ⊕ D t , W ⊕ D ∗ t ) . (2.18) While the entries in the ﬁrst ro w are clearly w ell-deﬁned and b ounded b etw een the resp ectiv e spaces (note that D t = H 1 0 (Ω) ∩ Dom( q 1 2 ) ∩ Dom( a 1 2 ) ⊂ W (2.19) is b oundedly embedded, see [27, Prop. 4.6]), the ones in the second row are deﬁned w eakly . More precisely , the op erators ∆ − q ∈ B ( W , W ∗ ) , a ∈ B ( D t , D ∗ t ) , are deﬁned by ((∆ − q ) u, v ) W ∗ ×W := − Z Ω ∇ u ( x ) ∇ v ( x )d x − Z Ω q ( x ) u ( x ) v ( x )d x, ( au, v ) D ∗ t ×D t := Z Ω a ( x ) u ( x ) v ( x )d x, (2.20) for all u and v in W or D t , resp ectiv ely (see App endix A for details on W ). In fact, the ab o v e deﬁned op erator − (∆ − q ) is nothing but the Riesz isomorphism J W : W → W ∗ . Hence, in particular, for u ∈ W and u ∗ ∈ W ∗ , ∥ (∆ − q ) u ∥ W ∗ = ∥ u ∥ W , ∥ (∆ − q ) − 1 u ∗ ∥ W = ∥ u ∗ ∥ W ∗ . (2.21) The op erator A is deﬁned as the maximal restriction of b A to H , in detail, A := b A| Dom( A ) , with Dom( A ) introduced in (2.7). Note that the ﬁrst row in (2.18) do es not con- tribute an y restriction to the domain of A since D t ⊂ W and we employ ed a triple D t ⊂ L 2 (Ω) ⊂ D ∗ t in the second space comp onent only . The adjoint of A has the exp ected structure. Prop osition 2.5. L et A b e as in (2.2) and (2.7) . Then the adjoint of A r e ads A ∗ =  0 − I − (∆ − q ) − a  , Dom( A ∗ ) =  F = ( f , g ) ∈ W × D t : (∆ − q ) f + ag ∈ L 2 (Ω)  . (2.22) Pr o of. W e prov e that the adjoin t of A can b e obtained as the restriction of the (adjoin t) distributional matrix b C :=  0 − I − (∆ − q ) − a  ∈ B ( W ⊕ D t , W ⊕ D ∗ t ) . W e denote the restriction of b C to the domain in (2.22) by C and we pro v e that C = A ∗ . Analogously as for A , it follows that C is m-dissipative and since also A ∗ is m-dissipative, it is suﬃcient to show only the inclusion C ⊂ A ∗ . T o this end, ﬁx ( f , g ) ∈ Dom( C ) and let ( u, v ) ∈ Dom( A ) be arbitrary . With (2.20) we compute ⟨C ( f , g ) , ( u, v ) ⟩ H = −⟨ g , u ⟩ W − ⟨ (∆ − q ) f + ag , v ⟩ L 2 = −⟨ g , u ⟩ W − ((∆ − q ) f , v ) W ∗ ×W − ( ag , v ) D ∗ t ×D t = ((∆ − q ) u, g ) W ∗ ×W + ⟨ v , f ⟩ W − ( av , g ) D ∗ t ×D t = ⟨ (∆ − q ) u − av , g ⟩ L 2 + ⟨ v , f ⟩ W = ⟨ ( f , g ) , A ( u, v ) ⟩ H . 10 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL It follows that ( f , g ) ∈ Dom( A ∗ ) and A ∗ ( f , g ) = C ( f , g ), i.e. that C ⊂ A ∗ . □ Finally , the resolven t of A is constructed in terms of b T − 1 λ as ( A − λ ) − 1 = −   λ − 1  I + b T − 1 λ (∆ − q )  b T − 1 λ b T − 1 λ (∆ − q ) λ b T − 1 λ   , (2.23) see [27, pro of of Thm. 2.8]. 2.4. Range of A and the space K . W e can characterize the range of A as Ran( A ) =  ( f , g ) ∈ W × L 2 (Ω) : ∃ ( u, v ) ∈ Dom( A ) , f = v , g = (∆ − q ) u − av  =  ( f , g ) ∈ D t × L 2 (Ω) : ∃ u ∈ W , g = (∆ − q ) u − af  =  ( f , g ) ∈ D t × L 2 (Ω) : af + g ∈ W ∗  , (2.24) where the bijectivit y of ∆ − q = − J W : W → W ∗ w as used in the last step. It turns out that the range of A is precisely the space K in troduced in (1.7) and which app ears in our main results. Prop osition 2.6. L et A b e as in (2.2) and (2.7) . Then the fol lowing claims hold. (i) Both A and A ∗ ar e inje ctive. (ii) Ran( A ) = K and we have ∥ F ∥ 2 H +   A − 1 F   2 H = ∥ F ∥ 2 K , F ∈ Ran( A ) . (2.25) Mor e over, the norm F = ( f , g ) 7→ ( ∥ f ∥ 2 D t + ∥ g ∥ 2 L 2 + ∥ af + g ∥ 2 W ∗ ) 1 2 is e quivalent to ∥·∥ K . In p articular, K ⊂ D t ⊕ L 2 (Ω) . (2.26) (iii) K is dense in H . Pr o of. (i) The claims follow from the injectivit y of ∆ − q on W (see (2.21) and Prop osition 2.5). (ii) F rom (2.24), w e hav e Ran( A ) ⊂ K . Since for F = ( f , g ) ∈ Ran( A ) A − 1 F =  (∆ − q ) − 1 ( g + af ) , f  , w e obtain by (2.21) that ∥ F ∥ 2 H +   A − 1 F   2 H = ∥ f ∥ 2 W + ∥ g ∥ 2 L 2 + ∥ g + af ∥ 2 W ∗ + ∥ f ∥ 2 L 2 = ∥ F ∥ 2 K . Moreo ver, using b T − 1 1 ∈ B ( D ∗ t , D t ) (see (2.17)), and the con tin uity of the em beddings W ∗ ⊂ D ∗ t and L 2 (Ω) ⊂ D ∗ t , we arriv e at ∥ a 1 2 f ∥ L 2 ≲ ∥ b T 1 f ∥ D ∗ t ≲ ∥ (∆ − q ) f ∥ W ∗ + ∥ af + g ∥ W ∗ + ∥ f ∥ L 2 + ∥ g ∥ L 2 ≲ ∥ F ∥ K . This shows the equiv alence of the norms for F ∈ Ran( A ). It remains to justify that K ⊂ Ran( A ). Let ( f , g ) ∈ K . W e hav e f ∈ H 1 0 (Ω) ∩ Dom( q 1 2 ), g ∈ L 2 (Ω), af ∈ L 1 loc (Ω) and af + g ∈ W ∗ ⊂ D ∗ t , so it is enough to show that a 1 2 f ∈ L 2 (Ω). T o this end, we write D ′ (Ω) ∋ ( − ∆ + q + a + 1) f = ( − ∆ + q ) f + ( af + g ) + f − g ∈ D ∗ t . Since b T 1 is bijectiv e betw een D t and D ∗ t , there exists h ∈ D t suc h that the iden tit y ( − ∆ + q + a + 1) f = b T 1 h = ( − ∆ + q + a + 1) h holds in D ′ (Ω). W e sho w below that f = h ∈ D t and hence a 1 2 f ∈ L 2 (Ω) as claimed. SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 11 Let u := f − h , then u ∈ H 1 0 (Ω) ∩ Dom( q 1 2 ), au ∈ L 1 loc (Ω) and ∆ u = ( q + a + 1) u ∈ L 1 loc (Ω) . Emplo ying Kato’s inequality (see e.g. [22, Sec. VI I.2.1] for details), we obtain ∆ | u | ≥ Re  u | u | ∆ u  = Re (( q + a + 1) | u | ) ≥ | u | in D ′ (Ω), i.e. for all 0 ≤ ϕ ∈ C ∞ c (Ω) ((1 − ∆) | u | , ϕ ) D ′ ×D ≤ 0 . Since u ∈ H 1 0 (Ω) ∩ Dom( q 1 2 ) ⊂ H 1 0 (Ω), also | u | ∈ H 1 0 (Ω). Moreo v er, there exists { ϕ k } ⊂ C ∞ c (Ω) with ϕ k ≥ 0, k ∈ N , such that ϕ k → | u | in H 1 0 (Ω) (see e.g. [22, Cor. VI.2.4, Thm. VI.3.6]). Hence ∥| u |∥ 2 H 1 = lim k →∞ ⟨| u | , ϕ k ⟩ H 1 = lim k →∞ ((1 − ∆) | u | , ϕ k ) D ′ ×D ≤ 0 , and we conclude that | u | = | f − h | = 0 a.e. in Ω. (iii) By (ii) and (i), we hav e K = Ran( A ) = Ker( A ∗ ) ⊥ = H . □ Corollary 2.7. L et A b e as in (2.2) and (2.7) . Then 0 ∈ ρ ( A ) if and only if ther e exists M > 0 such that ∥ a 1 2 f ∥ 2 L 2 + ∥ f ∥ 2 L 2 ≤ M  ∥∇ f ∥ 2 L 2 + ∥ q 1 2 f ∥ 2 L 2  , f ∈ C ∞ c (Ω); (2.27) in this c ase ( − 1 / M , 0] ⊂ ρ ( A ) . Mor e over, if 0 ∈ σ ( A ) , then 0 ∈ σ c ( A ) ⊂ σ e1 ( A ) . Pr o of. Assume that 0 ∈ ρ ( A ). It follows that Ran( A ) = K = H , th us the inclusions (2.19) and (2.26) yield that W = D t . Since the iden tit y map I D t →W : D t → W : f 7→ f is everywhere deﬁned and b ounded, the b ounded in v erse (or closed graph) theorem shows that I − 1 D t →W = I W →D t is b ounded. Hence there exists M > 0 such that for all f ∈ C ∞ c (Ω) ⊂ D t ∥∇ f ∥ 2 L 2 + ∥ q 1 2 f ∥ 2 L 2 + ∥ a 1 2 f ∥ 2 L 2 + ∥ f ∥ 2 L 2 ≤ M  ∥∇ f ∥ 2 L 2 + ∥ q 1 2 f ∥ 2 L 2  , and (2.27) follows. T o sho w the reverse implication, supp ose that (2.27) holds. It follows that for all λ ∈ ( − 1 / M , 0) t λ [ f ] ≥  1 − | λ | M  ∥∇ f ∥ 2 L 2 + ∥ q 1 2 f ∥ 2 L 2  + | λ | 2 ∥ f ∥ 2 L 2 ≳ ∥ f ∥ 2 D t , f ∈ C ∞ c (Ω) . Since C ∞ c (Ω) is dense in D t , the forms t λ , λ ∈ ( − 1 / M , 0), are co ercive on D t and so w e ha v e b T − 1 λ ∈ B ( D ∗ t , D t ). It follows that λ ∈ ρ ( A ) (see [27, Cor. 3.7, Sec. 4] for details). F or λ = 0, notice that (2.27) implies that (with equiv alen t norms) W = H 1 0 (Ω) ∩ Dom( q 1 2 ) = D t , (see (2.4)), and thus also W ∗ = D ∗ t . It is then straightforw ard to see from (2.24) that Ran( A ) = H , hence A is bijective and 0 ∈ ρ ( A ) b y the closed graph theorem. Finally , since Ran( A ) is dense in H and A is injective by Prop osition 2.6, it follo ws that if 0 ∈ σ ( A ), then 0 ∈ σ c ( A ). It is immediate from the deﬁnitions that σ c ( A ) ⊂ σ e1 ( A ). □ 2.5. Restriction of e t A semigroup to K . F or some estimates of Theorem 1.2, w e will use the restriction of the semigroup e t A to the space K . Deﬁne the op erator A K in K b y A K ⊂ A , Dom( A K ) = { U ∈ Dom( A ) ∩ K : A U ∈ K} = Dom( A ) ∩ Ran( A ) . (2.28) Prop osition 2.8. L et A b e as in (2.2) and (2.7) and A K as in (2.28) . 12 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL (i) We have ρ ( A ) ⊂ ρ ( A K ) and   ( A K − λ ) − 1   B ( K ) ≤   ( A − λ ) − 1   B ( H ) , λ ∈ ρ ( A ) . (2.29) (ii) F or t ≥ 0 we have e t A K ⊂ K and the r estriction (e t A | K ) t ≥ 0 deﬁnes a str ongly c ontinuous c ontr action semigr oup (e t A K ) t ≥ 0 on K with the densely deﬁne d gener ator A K . Pr o of. (i) Let λ ∈ ρ ( A ). If λ = 0 then H = K and the statemen t is clear. No w assume that λ  = 0. Since ( A − λ ) is injective, then so is ( A K − λ ). Let F ∈ K ⊂ H . Since ( A − λ ) is surjectiv e, there exists U ∈ Dom( A ) such that ( A − λ ) U = F . It follo ws that U = λ − 1 ( A U − F ) ∈ Ran( A ). Thus U ∈ Dom( A K ) and ( A K − λ ) U = F . This pro ves that ( A K − λ ) is surjective. Finally , for the b oundedness of the in verse w e estimate   ( A K − λ ) − 1 F   2 K =   A − 1 ( A − λ ) − 1 F   2 H +   ( A − λ ) − 1 F   2 H =   ( A − λ ) − 1 A − 1 F   2 H +   ( A − λ ) − 1 F   2 H ≤   ( A − λ ) − 1   2 B ( H )    A − 1 F   2 H + ∥ F ∥ 2 H  =   ( A − λ ) − 1   2 B ( H ) ∥ F ∥ 2 K , where (2.25) was used in the ﬁrst and last steps. (ii) First, we pro v e that (e t A | K ) t ≥ 0 deﬁnes a contraction semigroup on K . T o this end, let F ∈ Ran( A ) = K and let U ∈ Dom( A ) be such that F = A U . By [23, Lem. I I.1.3 (ii)], we ha ve e t A F = e t A A U = A e t A U ∈ K , so (e t A | K ) t ≥ 0 deﬁnes a semigroup on K . Similarly , A − 1 e t A F = A − 1 e t A A U = e t A U = e t A A − 1 F . Using (2.25) and that (e t A ) t ≥ 0 is contractiv e on H , we arriv e at   e t A F   2 K =   A − 1 e t A F   2 H +   e t A F   2 H =   e t A A − 1 F   2 H +   e t A F   2 H ≤   A − 1 F   2 H + ∥ F ∥ 2 H = ∥ F ∥ 2 K . W e similarly chec k that   e t A F − F   K → 0 as t → 0, and it follo ws that (e t A | K ) t ≥ 0 is a strongly con tinuous contraction semigroup on K . W e denote its generator by G and we sho w that G = A K . Notice ﬁrst that b y the Hille–Y osida theorem [23, Thm. I I.3.5], G is a densely deﬁned maximal dissipative operator in K (the dissipativity follows from the resol- v ent bound therein, see [33, Sec. V.3.10]). Since A K is also m-dissipative by (i), it is suﬃcient to pro v e one inclusion A K ⊂ G . F or U ∈ Dom( A K ) and F = A U , w e ha ve b y [23, Lem. I I.1.3 (iv)] e t G U − U = e t A U − U = A Z t 0 e s A U d s = Z t 0 e s A F d s, t ≥ 0 . It follows that U ∈ Dom( G ) and G U = F = A U = A K U , and hence A K ⊂ G . □ 3. Resol vent estima tes for low frequencies 3.1. Statemen ts for the low frequency resolven t estimates. The resolven t estimates for λ → 0 are obtained under the following assumption (weak er than the assumption used in Theorem 1.2). Assumption 3.1. Let Ω ⊂ R d b e op en and non-empt y and let 0 ≤ a, q ∈ L 1 loc (Ω). Supp ose that there are op en non-empt y sets Ω 1 , Ω 2 ⊂ Ω suc h that SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 13 (i) Ω 1 is a bounded region with Lipschitz contin uous boundary (see e.g. [10, Def. D.2.3.1, p. 815]), (ii) Ω 1 ∩ Ω 2 = ∅ ,  Ω 1 ∪ Ω 2  ◦ = Ω, | Ω \ (Ω 1 ∪ Ω 2 ) | = 0, (iii) ∥ 1 Ω 1 a ∥ L 1 > 0 and there exists a 0 > 0 suc h that a ( x ) ≥ a 0 a.e. in Ω 2 . W e remark that Assumption 3.1 is satisﬁed in the case when there exists a (non- empt y) ball B ( x 0 , r 0 ) ⊂ Ω with B ( x 0 , r 0 ) ⊂ Ω such that ∥ 1 B ( x 0 ,r 0 ) a ∥ L 1 > 0 and a ( x ) ≥ a 0 > 0 for a.e. x ∈ Ω \ B ( x 0 , r 0 ). F or an exterior domain Ω, we shall also explicitly consider damping co eﬃcien ts a which are un bounded at inﬁnit y . Assumption 3.2. Let Ω ⊂ R d b e an exterior domain in R d with a Lipschitz b oundary , let 0 ≤ a, q ∈ L 1 loc (Ω) and assume that there exist β > 0 and r 0 > 0 such that R d \ B (0 , r 0 ) ⊂ Ω and a ( x ) ≳ | x | β , a.e. | x | > r 0 . (3.1) Notice that Assumption 3.2 is stronger than Assumption 3.1, since under As- sumption 3.2 we get Assumption 3.1 by considering (for an y r > r 0 > 0) Ω 1 = B (0 , r ) ∩ Ω , Ω 2 = Ω \ B (0 , r ) . (3.2) W e set γ = ( 1 , under Assumption 3.1 , 2 2+ β , if Assumption 3.2 holds in addition . (3.3) Our main spectral result is the analysis of the resolven t of A near 0. T o formulate it we deﬁne the bounded op erators (see (2.26)) Π 1 : K → D t : ( f , g ) 7→ f , Π 2 : K → L 2 (Ω) : ( f , g ) 7→ g . (3.4) Theorem 3.3. L et A and A K b e as in (2.2) , (2.7) and (2.28) , r esp e ctively, and let Assumption 3.1 (or the str onger Assumption 3.2) hold. L et γ b e as in (3.3) . Then ther e exist τ 0 > 0 and C > 0 such that for λ ∈ τ 0 D + \ { 0 } , we have λ ∈ ρ ( A ) ∩ ρ ( A K ) and ∥ ( A − λ ) − 1 ∥ B ( H ) + ∥ ( A K − λ ) − 1 ∥ B ( K ) ≤ C | λ | − 1 , (3.5) ∥ ( A K − λ ) − 1 ∥ B ( K , H ) ≤ C, (3.6) ∥ Π 1 ( A K − λ ) − 1 ∥ B ( K ,L 2 ) ≤ C | λ | − γ 2 , (3.7) ∥ Π 1 ( A K − λ ) − 1 ∥ B ( K , D t ) ≤ C | λ | − 1 2 , (3.8) ∥ Π 2 ( A K − λ ) − 1 ∥ B ( K ,L 2 ) ≤ C, (3.9) ∥ Π 2 ( A K − λ ) − 2 ∥ B ( K ,L 2 ) ≤ C | λ | − γ 2 . (3.10) (In (3.6) it is understo o d that ( A K − λ ) − 1 is c omp ose d on the left with the emb e dding of K into H , and in (3.7) that D t is emb e dde d in L 2 (Ω) .) In the case of our main interest, i.e. a strong damping for which (1.6) is not satisﬁed, zero b elongs to the spectrum of A (see Corollary 2.7). Thus the rate in (3.5) ab o v e cannot b e impro v ed and we cannot hav e uniform decay for e t A (see Prop osition 1.1). Ho w ever, we still ha v e some (polynomial) decay in some suitable top ology and the rates in Theorem 1.2 reﬂect the nature of the singularit y of the resolv ent ( A − λ ) − 1 around zero. Details are given in Section 5. Theorem 3.3 is pro v ed b elo w. The main step is the follo wing estimate on the form t λ for λ ∈ D + near zero. 14 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL Prop osition 3.4. L et Assumption 3.1 (or the str onger Assumption 3.2) hold and let the form t λ b e as in (2.10) . Then as λ → 0 in D + \ { 0 } | t λ [ u ] | ≳ ∥∇ u ∥ 2 L 2 + ∥ q 1 2 u ∥ 2 L 2 + | λ |∥ a 1 2 u ∥ 2 L 2 + | λ | γ ∥ u ∥ 2 L 2 , u ∈ D t , (3.11) wher e γ is as in (3.3) . In Section 3.2 w e deduce Theorem 3.3 from Proposition 3.4. Then w e pro vide a pro of of Prop osition 3.4 in Section 3.3. Remark 3.5. (i) If Ω and q are such that (2.3) holds, one can easily impro v e (3.11) to γ = 0, formally corresp onding to β = ∞ (see (3.21)). This will also impro ve (1.12) and (1.13) accordingly . (ii) F or q = 0 and a ( x ) = | x | β , x ∈ R d , a s caling argument shows that the rate γ = 2 / (2 + β ) in (3.11) is in fact optimal. (iii) The conditions in Assumption 3.1 can b e further relaxed. Consider the example a ( x, y ) = x 2 y 2 , ( x, y ) ∈ R 2 , for whic h the self-adjoint op erator − ∆ + x 2 y 2 in L 2 ( R 2 ) has compact resolv ent and the low est eigenv alue is p ositive (see e.g. [50]). By a scaling argumen t applied on the right-hand side of (3.21), one obtains that (3.11) holds with γ = 1 / 3 (lik e for a ( x, y ) = | ( x, y ) | 4 ). Note that a (0 , y ) = a ( x, 0) = 0, so Assumption 3.1 is not satisﬁed in this example. 3.2. The resolven t norm of A near zero. Assume that Prop osition 3.4 holds. The inequality (3.11) allows us to estimate the norm of the distributional Sc h ur com- plemen t b T λ b et w een suitable spaces, which leads to an estimate of the F rob enius– Sc hur factorization (2.23) of the resolven t. Lemma 3.6. L et b T λ b e deﬁne d as in (2.16) and let Assumption 3.1 or the str onger Assumption 3.2 hold. Then ther e exists τ 0 > 0 such that b T − 1 λ : D ∗ t → D t is b ounde d for λ ∈ τ 0 D + \ { 0 } and, as λ → 0 , ∥ b T − 1 λ ∥ B ( W ∗ , W ) ≲ 1 , ∥ b T − 1 λ ∥ B ( W ∗ ,L 2 ) ≲ | λ | − γ 2 , ∥ b T − 1 λ ∥ B ( L 2 , W ) ≲ | λ | − γ 2 , ∥ b T − 1 λ ∥ B ( L 2 ,L 2 ) ≲ | λ | − γ , ∥ b T − 1 λ ∥ B ( W ∗ , D t ) ≲ | λ | − 1 2 , ∥ b T − 1 λ ∥ B ( L 2 , D t ) ≲ | λ | − γ +1 2 , (3.12) wher e γ is as in (3.3) . Pr o of. Note ﬁrst that by (3.11) there exists τ 0 > 0 suc h that for all 0  = λ ∈ τ 0 D + w e ha ve | t λ [ u ] | ≳ | λ |∥ u ∥ 2 D t , and thus b T − 1 λ : D ∗ t → D t is b ounded. Since D t is a (dense and b oundedly em bedded) subspace of W and L 2 (Ω), it follows that L 2 (Ω) and W ∗ are (dense and b oundedly embedded) subspaces of D ∗ t . Th us one can indeed view b T − 1 λ as an op erator b etw e en the v arious spaces as in (3.12), where the resp ectiv e restrictions are not written explicitly as they are clear from the context. W e show the second inequalit y in the ﬁrst line of (3.12), the remaining ones are justiﬁed analogously . By Prop osition 3.4 we ha v e, as λ → 0 in D + \ { 0 } , | ( b T λ u, u ) D ∗ t ×D t | ≳ | λ | γ 2 ∥ u ∥ W ∥ u ∥ L 2 , u ∈ D t . (3.13) Hence, using (3.13) for u = b T − 1 λ ϕ with ϕ ∈ W ∗ ∥ b T − 1 λ ∥ B ( W ∗ ,L 2 ) = sup 0  = φ ∈W ∗ ∥ b T − 1 λ ϕ ∥ L 2 ∥ ϕ ∥ W ∗ ≲ | λ | − γ 2 sup 0  = φ ∈W ∗ | ( ϕ, b T − 1 λ ϕ ) W ∗ ×W | ∥ ϕ ∥ W ∗ ∥ b T − 1 λ ϕ ∥ W ≲ | λ | − γ 2 ; notice that the pairing in D ∗ t × D t can b e written in the duality W ∗ × W since ϕ ∈ W ∗ and b T − 1 λ ϕ ∈ D t ⊂ W . □ SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 15 Pr o of of The or em 3.3. W e pro v e ﬁrst (3.5), based on the estimates in Lemma 3.6 and the representation of the resolven t in (2.23). Notice that τ 0 D + \ { 0 } ⊂ ρ ( A ) ⊂ ρ ( A K ) by the sp ectral equiv alence (2.12) and the bounded inv ertibility of T λ for λ ∈ τ 0 D + \ { 0 } in Lemma 3.6 (see also Prop osition 2.8). F or any F = ( f , g ) ∈ H w e hav e ∥ ( A − λ ) − 1 F ∥ H ≲ | λ | − 1  ∥ f ∥ W + ∥ b T − 1 λ (∆ − q ) f ∥ W  + ∥ b T − 1 λ g ∥ W + ∥ b T − 1 λ (∆ − q ) f ∥ L 2 + | λ |∥ b T − 1 λ g ∥ L 2 ≲ | λ | − 1 ∥ f ∥ W + | λ | − 1 ∥ b T − 1 λ ∥ B ( W ∗ , W ) ∥ (∆ − q ) ∥ B ( W , W ∗ ) ∥ f ∥ W + ∥ b T − 1 λ ∥ B ( W ∗ ,L 2 ) ∥ (∆ − q ) ∥ B ( W , W ∗ ) ∥ f ∥ W + ∥ b T − 1 λ ∥ B ( L 2 , W ) ∥ g ∥ L 2 + | λ |∥ b T − 1 λ ∥ B ( L 2 ) ∥ g ∥ L 2 . Emplo ying Lemma 3.6, (3.3) and (2.21), we obtain ∥ ( A − λ ) − 1 F ∥ H ≲ | λ | − 1 ∥ F ∥ H , i.e. (3.5) holds for A . Then the estimate for A K in (3.5) follows from (2.29). Notice that if we only consider the second row in (2.23) we get ∥ Π 2 ( A − λ ) − 1 F ∥ L 2 ≲ | λ | − γ 2 ∥ F ∥ H . (3.14) T o justify the remaining inequalities, w e ﬁrst rearrange the form ula (2.23) for ( A − λ ) − 1 F when F = ( f , g ) ∈ K = Ran( A ) (see (2.24) and Prop osition 2.6). Since f ∈ D t implies af ∈ D ∗ t , we ha v e b T − 1 λ (∆ − q ) f = b T − 1 λ  (∆ − q ) f − λaf − λ 2 f + λaf + λ 2 f  = − f + λ b T − 1 λ ( af + λf ) . Hence, ( A − λ ) − 1 F = − b T − 1 λ ( af + g ) + λ b T − 1 λ f λ b T − 1 λ ( af + g ) + λ 2 b T − 1 λ f − f ! . W e set U = ( u, v ) = ( A − λ ) − 1 F . Since af + g ∈ W ∗ and f ∈ L 2 (Ω), by (3.12) w e arriv e at (with ι, κ ∈ { 0 , 1 } ) ∥ u ∥ W + ι ∥ u ∥ L 2 + κ ∥ a 1 2 u ∥ L 2 ≲  ∥ b T − 1 λ ∥ B ( W ∗ , W ) + ι ∥ b T − 1 λ ∥ B ( W ∗ ,L 2 ) + κ ∥ b T − 1 λ ∥ B ( W ∗ , D t )  ∥ af + g ∥ W ∗ + | λ |  ∥ b T − 1 λ ∥ B ( L 2 , W ) + ι ∥ b T − 1 λ ∥ B ( L 2 ,L 2 ) + κ ∥ b T − 1 λ ∥ B ( L 2 , D t )  ∥ f ∥ L 2 ≲  1 + ι | λ | − γ 2 + κ | λ | − 1 2  ∥ af + g ∥ W ∗ +  | λ | 1 − γ 2 + ι | λ | 1 − γ + κ | λ | 1 − γ +1 2  ∥ f ∥ L 2 ≲  1 + ι | λ | − γ 2 + κ | λ | − 1 2  ∥ F ∥ K and ∥ v ∥ ≲ | λ |∥ b T − 1 λ ∥ B ( W ∗ ,L 2 ) ∥ af + g ∥ W ∗ + | λ | 2 ∥ b T − 1 λ ∥ B ( L 2 ) ∥ f ∥ L 2 + ∥ f ∥ L 2 ≲ ∥ f ∥ L 2 + ∥ af + g ∥ W ∗ ≲ ∥ F ∥ K , (see (1.8)). The inequalities (3.6), (3.7) and (3.8) follow b y taking ι = κ = 0, ι = 1 and κ = 0, and ι = κ = 1, resp ectively . Finally , (3.9) follows from (3.6) since ∥ Π 2 ∥ B ( H ,L 2 ) = 1 and (3.10) follows from (3.6) and (3.14). □ 3.3. Lo w frequency estimate for the asso ciated quadratic form. In the rest of the section we pro v e Prop osition 3.4. The pro of is based on a classical Neumann-brac keting argument (see [46, Chap. XI I I.15]) and asymptotic p erturba- tion theory for a family of self-adjoin t op erators in the sense of quadratic forms (see [33, Chap. VI I I]). The main ingredien t is the following lo wer bound. 16 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL Lemma 3.7. L et Assumption 3.1 or Assumption 3.2 hold. Then as b → 0 + ∥∇ u ∥ 2 L 2 + b ∥ a 1 2 u ∥ 2 L 2 ≳ b γ ∥ u ∥ 2 L 2 , u ∈ H 1 0 (Ω) ∩ Dom( a 1 2 ) , (3.15) wher e γ is as in (3.3) . Pr o of. Let u ∈ H 1 0 (Ω) ∩ Dom( a 1 2 ) be arbitrary . W e split the problem into its parts on Ω 1 and Ω 2 as in Assumption 3.1. More precisely , ∥∇ u ∥ 2 L 2 (Ω) + b ∥ a 1 2 u ∥ 2 L 2 (Ω) = 2 X j =1  ∥∇ u j ∥ 2 L 2 (Ω j ) + b ∥ a 1 2 u j ∥ 2 L 2 (Ω j )  , (3.16) where w e write u j := u | Ω j for j = 1 , 2. W e thus ma y pro v e (3.15) for eac h summand separately . • Assumption 3.1: Let us ﬁrst consider the case when only Assumption 3.1 holds. F or the j = 2 summand, we clearly ha v e by Assumption 3.1 (iii) ∥∇ u 2 ∥ 2 L 2 (Ω 2 ) + b ∥ a 1 2 u 2 ∥ 2 L 2 (Ω 2 ) ≥ a 0 b ∥ u 2 ∥ 2 L 2 (Ω 2 ) . (3.17) F or j = 1, w e use a p erturbative argument in L 2 (Ω 1 ). Let H 0 denote the Neu- mann Laplacian on Ω 1 , i.e. the self-adjoint operator asso ciated with the quadratic form h 0 [ v ] := ∥∇ v ∥ 2 L 2 (Ω 1 ) , Dom( h 0 ) := H 1 (Ω 1 ) , and consider the form p erturbation a [ v ] := ∥ a 1 2 v ∥ 2 L 2 (Ω 1 ) , Dom( a ) := Dom( a 1 2 | Ω 1 ) . Using [33, Thm. VII I.4.9] we derive an expansion for the low est eigenv alue λ 0 ( b ) of the op erator H b asso ciated with the form h b := h 0 + b a in the regime b → 0 + . Note that both h 0 and a are densely deﬁned, closed and non-negativ e. Since Ω 1 has Lipsc hitz boundary , the restrictions of functions in C ∞ c ( R d ) to Ω 1 are dense in H 1 (Ω 1 ) (see [10, Thm. 8.10.7]). Since these functions are also contained in Dom( h b ) = H 1 (Ω 1 ) ∩ Dom( a 1 2 | Ω 1 ) , the latter is dense in Dom( h 0 ). Hence, the conditions in [33, p. 464] hold with t = h 0 and t ( 1 ) = a . Moreo v er, by [33, Thms. VI II.3.11, VI II.3.15], the operators H b con- v erge to H 0 strongly in the generalized sense as b → 0 + . Since Ω 1 has Lipsc hitz b oundary , the limit H 0 has compact resolven t (see e.g. [10, Thm. 8.11.4, p. 614]), while its low est eigenv alue λ 0 (0) = 0 is simple and stable (in the sense of the deﬁni- tion in [33, p. 437]) with constant eigenfunction. With P 0 denoting the orthogonal pro jection on the corresp onding eigenspace span { 1 } , recalling that a ∈ L 1 loc (Ω) by assumption, we ha v e Ran( P 0 ) ⊂ Dom( a 1 2 | Ω 1 ) = Dom( t ( 1 ) ) . By the previous paragraph, the assumptions of [33, Thm. VII I.4.9] are satisﬁed. Hence, the low est eigen v alue of H b admits the expansion λ 0 ( b ) = b | Ω 1 | − 1 ∥ a | Ω 1 ∥ L 1 (Ω 1 ) + o ( b ) , b → 0 + . Considering Assumption 3.1 (iii), it th us follows that h b [ v ] ≳ b ∥ v ∥ 2 L 2 (Ω 2 ) , v ∈ Dom( h b ) . (3.18) No w (3.15) with γ = 1 follo ws from (3.16), (3.17) and (3.18) with v = u 1 ∈ Dom( h b ). • Assumption 3.2: Let Assumption 3.2 hold in addition, where Ω 1 and Ω 2 are as in (3.2) with r ≡ r ( b ) := b − 1 2+ β . F or the summand j = 2 in (3.16), we apply (3.1) to ﬁnd ∥∇ u 2 ∥ 2 L 2 (Ω 2 ) + b ∥ a 1 2 u 2 ∥ 2 L 2 (Ω 2 ) ≥ br β ∥ u 2 ∥ 2 L 2 (Ω 2 ) = b 2 2+ β ∥ u 2 ∥ 2 L 2 (Ω 2 ) . (3.19) SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 17 T o treat the summand j = 1, let v denote the zero extension of u to R d , i.e. v ∈ H 1 ( R d ) with v ( x ) = 0 on R d \ Ω. Note that, for b small enough, w e ha v e r ≥ 2 r 0 and thus B := B (0 , 1) \ B (0 , 1 2 ) ⊂ B (0 , 1) \ B (0 , r 0 r ) ⊂ r − 1 Ω 1 . Setting w ( y ) = r d 2 v ( ry ) and using (3.1), we obtain the follo wing low er estimate ∥∇ u 1 ∥ 2 L 2 (Ω 1 ) + b ∥ a 1 2 u 1 ∥ 2 L 2 (Ω 1 ) = Z B (0 ,r ) |∇ v ( x ) | 2 d x + b Z Ω 1 a ( x ) | v ( x ) | 2 d x = r − 2 Z B (0 , 1) |∇ w ( y ) | 2 d y + b Z r − 1 Ω 1 a ( r y ) | w ( y ) | 2 d y ≳ r − 2 Z B (0 , 1) |∇ w ( y ) | 2 d y + Z B | y | β | w ( y ) | 2 d y ! . It is easy to see that the Neumann realization of − ∆ + 1 B | y | β in L 2 ( B (0 , 1)), i.e. the self-adjoin t op erator asso ciated with the form s [ w ] := ∥∇ w ∥ 2 L 2 ( B (1 , 0)) + ∥ 1 B | y | β 2 w ∥ 2 L 2 ( B (1 , 0)) , Dom( s ) := H 1 ( B (1 , 0)) , has compact resolven t and a p ositiv e lo west eigen v alue. This further implies ∥∇ u 1 ∥ 2 L 2 (Ω 1 ) + b ∥ a 1 2 u 1 ∥ 2 L 2 (Ω 1 ) ≳ r − 2 ∥ w ∥ 2 L 2 ( B (0 , 1)) = b 2 2+ β ∥ u 1 ∥ 2 L 2 (Ω 1 ) . (3.20) Putting together (3.16), (3.19) and (3.20), the claim (3.15) follows with γ = 2 / (2 + β ). □ Pr o of of Pr op osition 3.4. F or an y u ∈ D t and λ ∈ D + \ { 0 } , setting b := | λ | > 0 it can b e readily veriﬁed that | t λ [ u ] | ≥ | Re(e − i 2 arg λ t λ [ u ]) | ≥ √ 2 2  ∥∇ u ∥ 2 L 2 + ∥ q 1 2 u ∥ 2 L 2 + b ∥ a 1 2 u ∥ 2 L 2  − b 2 ∥ u ∥ 2 L 2 . (3.21) The claim now follo ws from Lemma 3.7. □ 4. Resol vent estima tes for high frequencies It is for high frequencies that w e add the assumption a ( x ) ≥ a 0 > 0 a.e. in Ω in the statement of Theorem 1.2. Under this assumption, implying in particular the GCC, we prov e uniform b oundedness of the resolven t of A on the imaginary axis a wa y from zero. The challenge in the pro of is the minimal regularity of the coeﬃ - cien ts. Some cases with non-uniformly p ositiv e damping are discussed in Section 7. Prop osition 4.1. L et A and A K b e as in (2.2) , (2.7) and (2.28) , r esp e ctively. Assume that ther e exists a 0 > 0 such that a ( x ) ≥ a 0 a.e. in Ω . Then C + \{ 0 } ⊂ ρ ( A ) and for al l ε > 0 it holds that sup λ ∈ C + \ ε D ∥ ( A − λ ) − 1 ∥ B ( H ) < ∞ , sup λ ∈ C + \ ε D ∥ ( A K − λ ) − 1 ∥ B ( K ) < ∞ . (4.1) The proof of Prop osition 4.1 relies on a simple resolv ent estimate for T λ with λ ∈ i R together with the following lemma. The latter is an extension of the argumen t in [5, Pro of of Thm. 3.5] and establishes the respective resolven t estimates for A (without assuming uniform p ositivity of the damping). 18 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL Lemma 4.2. L et A and T λ b e as in (2.2) , (2.7) and (2.11) , r esp e ctively. Assume that for some b 0 > 0 we have σ ( A ) ∩ i R ⊂ i( − b 0 , b 0 ) . Then ther e exists C 0 > 0 such that ∥ ( A − i b ) − 1 ∥ B ( H ) ≤ C 0  1 | b | + | b |∥ T − 1 i b ∥ B ( L 2 )  , | b | ≥ b 0 . Pr o of. W e recall that by (2.12) and Lemma 2.4 the operator b T i b deﬁned as in (2.16) is b oundedly in v ertible for | b | ≥ b 0 and ( A − i b ) − 1 is given by (2.23). Let F = ( f , g ) ∈ H . By (2.21) we ha v e ∥ ( A − i b ) − 1 F ∥ H ≲ | b | − 1  ∥ f ∥ W + ∥ b T − 1 i b (∆ − q ) f ∥ W  + ∥ b T − 1 i b g ∥ W + ∥ b T − 1 i b (∆ − q ) f ∥ L 2 + | b |∥ b T − 1 i b g ∥ L 2 ≲ | b | − 1  1 + ∥ b T − 1 i b ∥ B ( W ∗ , W )  ∥ f ∥ W + ∥ b T − 1 i b ∥ B ( W ∗ ,L 2 ) ∥ f ∥ W + ∥ T − 1 i b ∥ B ( L 2 , W ) ∥ g ∥ L 2 + | b |∥ T − 1 i b ∥ B ( L 2 ) ∥ g ∥ L 2 . (4.2) F or u ∈ D t and b ∈ R w e hav e ∥ u ∥ 2 W = Re( t i b [ u ]) + b 2 ∥ u ∥ 2 L 2 ≤ | t i b [ u ] | + b 2 ∥ u ∥ 2 L 2 . F or v ∈ L 2 (Ω) we can use this with u = T − 1 i b v ∈ Dom( T i b ) to arrive at ∥ T − 1 i b v ∥ 2 W ≲  ∥ T − 1 i b ∥ B ( L 2 ) + b 2 ∥ T − 1 i b ∥ 2 B ( L 2 )  ∥ v ∥ 2 L 2 . Hence, as | b | → ∞ , ∥ T − 1 i b ∥ B ( L 2 , W ) ≲ q ∥ T − 1 i b ∥ B ( L 2 ) + | b |∥ T − 1 i b ∥ B ( L 2 ) ≲ 1 | b | + | b |∥ T − 1 i b ∥ B ( L 2 ) . (4.3) Next we estimate for arbitrary ϕ ∈ W ∗ that ∥ b T − 1 i b ϕ ∥ L 2 = sup 0  = f ∈ L 2 (Ω) |⟨ f , b T − 1 i b ϕ ⟩ L 2 | ∥ f ∥ L 2 = sup 0  = f ∈ L 2 (Ω) | t − i b [ b T − 1 − i b f , b T − 1 i b ϕ ] | ∥ f ∥ L 2 = sup 0  = f ∈ L 2 (Ω) | t i b [ b T − 1 i b ϕ, b T − 1 − i b f ] | ∥ f ∥ L 2 = sup 0  = f ∈ L 2 (Ω) | ( ϕ, b T − 1 − i b f ) W ∗ ×W | ∥ f ∥ L 2 ≤ ∥ ϕ ∥ W ∗ sup 0  = f ∈ L 2 (Ω) ∥ b T − 1 − i b f ∥ W ∥ f ∥ L 2 ≤ ∥ ϕ ∥ W ∗ ∥ T − 1 − i b ∥ B ( L 2 , W ) and hence ∥ b T − 1 i b ∥ B ( W ∗ ,L 2 ) ≤ ∥ T − 1 − i b ∥ B ( L 2 , W ) ≲ 1 | b | + | b |∥ T − 1 i b ∥ B ( L 2 ) , | b | → ∞ . (4.4) Finally , w e estimate the remaining term ∥ b T − 1 i b ∥ B ( W ∗ , W ) . F or distinct sp ectral parameters λ, µ ∈ C \ ( −∞ , 0] with 0 ∈ ρ ( T λ ) ∩ ρ ( T µ ), the second resolven t identit y leads to b T − 1 λ = b T − 1 µ + ( µ − λ ) b T − 1 µ ( a + λ + µ ) b T − 1 λ = µ λ b T − 1 µ + µ ( µ − λ ) b T − 1 µ b T − 1 λ + µ − λ λ b T − 1 µ (∆ − q ) b T − 1 λ . Note that there are no issues with the domains of the in v olved op erators since b T − 1 λ , b T − 1 µ ∈ B ( D ∗ t , D t ) and a, (∆ − q ) ∈ B ( D t , D ∗ t ). F urther manipulations yield  b T − 1 µ (∆ − q ) + λ λ − µ  b T − 1 λ = µ λ − µ b T − 1 µ − λµ b T − 1 µ b T − 1 λ . (4.5) SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 19 W e restrict this identit y to W ∗ and select λ = i b , µ = b . The next step is to inv ert the op erator in parentheses on the left hand side, i.e. B − i i − 1 where B := I D t →W b T − 1 b I W ∗ →D ∗ t J W ∈ B ( W ); recall that J W = − (∆ − q ) : W → W ∗ is the Riesz isomorphism (see (2.20)). F or u, v ∈ W we deriv e ⟨ u, B v ⟩ W = ( J W u, I D t →W b T − 1 b I W ∗ →D ∗ t J W v ) W ∗ ×W = ( I W ∗ →D ∗ t J W u, b T − 1 b I W ∗ →D ∗ t J W v ) D ∗ t ×D t = t b [ b T − 1 b I W ∗ →D ∗ t J W u, b T − 1 b I W ∗ →D ∗ t J W v ] Due to the symmetry of t b , it follo ws that B is symmetric and hence self-adjoin t. Then  B − i / (i − 1)  is b oundedly in v ertible and       B − i i − 1  − 1      B ( W ) ≤ 2 . (4.6) Observ e that it follows from t b [ u ] = ∥∇ u ∥ 2 L 2 + ∥ q 1 2 u ∥ 2 L 2 + b ∥ a 1 2 u ∥ 2 L 2 + b 2 ∥ u ∥ 2 L 2 , u ∈ D t , that (see the pro of of Lemma 3.6 for details on analogous arguments) ∥ b T − 1 b ∥ B ( W ∗ , W ) ≤ 1 , ∥ T − 1 b ∥ B ( L 2 , W ) ≲ 1 | b | , | b | → ∞ . (4.7) In summary , we ha ve from (4.5), (4.6), (4.7) and (4.4) that ∥ b T − 1 i b ∥ B ( W ∗ , W ) ≤ 2  ∥ b T − 1 b ∥ B ( W ∗ , W ) + b 2 ∥ T − 1 b ∥ B ( L 2 , W ) ∥ b T − 1 i b ∥ B ( W ∗ ,L 2 )  ≲ 1 + b 2 1 | b |  1 | b | + | b |∥ T − 1 i b ∥ B ( L 2 )  , b → ∞ . (4.8) Th us the lemma follows from (4.2), (4.3), (4.4) and (4.8). □ Pr o of of Pr op osition 4.1. F or the ﬁrst claim C + \ { 0 } ⊂ ρ ( A ) , see Theorem 2.3 (ii). F or u ∈ D t w e hav e | t i b [ u ] | ≥ | Im t i b [ u ] | = | b |∥ a 1 2 u ∥ 2 L 2 ≥ | b | a 0 ∥ u ∥ 2 L 2 , so ∥ T − 1 i b ∥ B ( L 2 ) ≲ | b | − 1 for 0  = b ∈ R and the ﬁrst part of (4.1) is obtained b y Lemma 4.2. The second estimate then follows from (2.29). □ 5. Time deca y In this section w e recall ho w in general one can con v ert resolv ent estimates in to time deca y for the corresponding semigroup, and in particular we prov e that Theorem 3.3 and Prop osition 4.1 imply Theorem 1.2. 5.1. Pro of of the main results. W e b egin with Prop osition 1.1. Pr o of of Pr op osition 1.1. By Corollary 2.7, (1.6) holds if and only if 0 ∈ ρ ( A ). First, if (1.5) holds, then 0 ∈ ρ ( A ) by [23, Thm. I I.3.8]. Conv ersely , assuming uni- form p ositivit y of a , it follo ws from 0 ∈ ρ ( A ) and Proposition 4.1 that the resolv en t of A is well-deﬁned and uniformly b ounded on the imaginary axis. Hence, (1.5) holds by the Gearhart–Pr¨ uss theorem [23, Thm. V.1.11]. □ 20 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL Remark 5.1. If the op erator norm of the semigroup dec a ys in time, the deca y is necessarily exponential (see [23, Prop. V.1.2]). Hence, when 0 ∈ σ ( A ) we cannot ha ve a norm decay of the solutions U ( t ) = e t A F whic h is uniform with resp ect to the initial v alue F ∈ H , i.e. an estimate of the type ∀ t ≥ 0 , ∀ F ∈ H : ∥ U ( t ) ∥ H ≤ ρ ( t ) ∥ F ∥ H with a p ositive function ρ decaying at inﬁnity . T o obtain the uniform estimates in Theorem 1.2, one th us restricts to a subspace of initial v alues F ∈ K . Nev er- theless, the energy of the solution decays to 0 for any initial v alue. This follows from the b oundedness of the semigroup (e t A ) t ≥ 0 , the estimates (1.9) and (1.10) in Theorem 1.2 and the density of K in H . No w w e turn to the pro of of the main Theorem 1.2. F or these estimates, w e use the following abstract result. Theorem 5.2. L et K b e a Hilb ert sp ac e and let A b e the gener ator of a str ongly c ontinuous semigr oup on K . L et X and Y b e norme d ve ctor sp ac es. L et T X ∈ B ( X , K ) and T Y ∈ B ( K , Y ) . Supp ose that the fol lowing two assumptions hold. (i) Ther e exists τ 0 ≥ 1 such that C + \ i( − τ 0 , τ 0 ) ⊂ ρ ( A ) , and we have sup λ ∈ C + \ τ 0 D   ( A − λ ) − 1   B ( K ) < ∞ . (5.1) (ii) Ther e exist m ∈ N 0 , κ ∈ [0 , 1) and C 0 > 0 such that for al l j ∈ ( { m, m + 1 , m + 2 } if κ = 0 , { m, m + 1 } if κ  = 0 , and λ ∈ τ 0 D + , we have   T Y ( A − λ ) − 1 − j T X   B ( X , Y ) ≤ C 0 | λ | m − κ − j . (5.2) Then ther e exists C > 0 such that for al l t ≥ 0 we have   T Y e tA T X   B ( X , Y ) ≤ C ⟨ t ⟩ − m − 1+ κ . (5.3) The pro of of Theorem 5.2 is postp oned to Section 5.2. No w we use Theorem 5.2 to prov e Theorem 1.2. Pr o of of The or em 1.2. W e apply Theorem 5.2 with A = A K , K = X = K and T X = I K . The assumption (5.1) holds by Proposition 4.1. W e b egin with (1.9). By (3.5) and (3.6) in Theorem 3.3, the estimates (5.2) hold with Y = H , T Y = I K→H and m = κ = 0. Then Theorem 5.2 giv es   e t A   B ( K , H ) ≲ ⟨ t ⟩ − 1 . This prov es in particular (1.9) (as well as an estimate for ∥ ∂ t u ( t ) ∥ L 2 whic h is not go od enough). Next we prov e (1.10) and (1.12) sim ultaneously . Indeed, setting Y = L 2 (Ω), T Y = Π 2 , m = 1 and κ = γ 2 (see (3.3) and (3.4)), it follo ws from (3.5), (3.9) and (3.10) in Theorem 3.3 that   Π 2 e t A   B ( K ,L 2 ) ≲ ⟨ t ⟩ γ 2 − 2 . F or (1.11) w e c ho ose Y = D t , T Y = Π 1 : K → D t (see (3.4)), m = 0 and κ = 1 2 . The claim then follows since (3.8) and (3.5) give   Π 1 e t A   B ( K , D t ) ≲ ⟨ t ⟩ − 1 2 . SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 21 Finally , for (1.13) we use the estimates (3.7), (3.5) to apply Theorem 5.2 with Y = L 2 (Ω), T Y = I D t → L 2 Π 1 : K → L 2 (Ω), m = 0 and κ = γ 2 . This yields the ﬁnal claim by   Π 1 e t A   B ( K ,L 2 ) ≲ ⟨ t ⟩ γ 2 − 1 . □ Remark 5.3. Estimates of the solution for initial conditions in the range of the generator hav e b een prov ed in general settings. With the resolv ent estimates (3.5) and (4.1), we can apply [9, Thm. 7.6] to our b ounded (con tractiv e) C 0 -semigroup (e t A ) t ≥ 0 in H , which giv es ∥ e t A A ( A − 1) − 1 ∥ B ( H ) = O  t − 1  , t → ∞ . Then (1.9) follows with Proposition 2.6, since for F ∈ K we get ∥ e t A F ∥ H ≲ ⟨ t ⟩ − 1 ∥ ( A − 1) A − 1 F ∥ H ≲ ⟨ t ⟩ − 1 ∥ F ∥ K . (5.4) This how ever do es not giv e (1.10) for ∥ ∂ t u ( t ) ∥ L 2 . 5.2. F rom resolven t estimates to time decay. In this paragraph we prov e Theorem 5.2. The proof is inspired by standard ideas (see the proof of the Gearhart– Pr ¨ uss theorem for the contribution of high frequencies and [11, Prop. 5.3] for the con tribution of low frequencies). Pr o of of The or em 5.2. • Appr oximating e tA in terms of r esolvent: Let M ≥ 1 and ω ≥ 0 be suc h that ∀ t ≥ 0 : ∥ e tA ∥ B ( K ) ≤ M e tω , (see [23, Prop. I.5.5] and notice that if we can tak e ω < 0 then the result is clear). W e c hoose some ﬁxed µ > ω . Let ϕ ∈ X and recall (see [23, Thm. I I.1.10 (ii)]) that for all τ ∈ R we ha v e in K ( A − ( µ + i τ )) − 1 T X ϕ = − Z ∞ 0 e t ( A − ( µ +i τ )) T X ϕ d t. (5.5) Th us τ 7→ ( A − ( µ + i τ )) − 1 T X ϕ is the F ourier transform of t 7→ − 1 R + ( t )e − tµ e tA T X ϕ (whic h b elongs to L 1 ( R ; K ) ∩ L 2 ( R ; K )). F or R > 0 we set e U R ( t ) = − 1 2 π Z R − R e i tτ ( A − ( µ + i τ )) − 1 d τ ∈ B ( K ) . Then it follows from [49, Rem. 5.1.5] that Z ∞ 0 ∥ T Y  e t ( A − µ ) − e U R ( t )  T X ϕ ∥ 2 Y d t − − − − → R →∞ 0 . (5.6) • Deriving suﬃcient c ondition for (5.3) : W e set U R ( t ) = e tµ e U R ( t ) = − 1 2i π Z Γ R e tλ ( A − λ ) − 1 d λ, where Γ R is the line segment joining µ − i R to µ + i R . W e sho w b elo w that if there exists a constant C > 0 such that ∀ ϕ ∈ X , ∀ t ≥ 0 : lim sup R →∞ ∥ T Y U R ( t ) T X ϕ ∥ Y ≤ C ⟨ t ⟩ − m − 1+ κ ∥ ϕ ∥ X , (5.7) then (5.3) holds with this constant C . Indeed, assume for con tradiction that there exist t 0 > 0 and ϕ ∈ X suc h that η = ∥ T Y e t 0 A T X ϕ ∥ Y − C ⟨ t 0 ⟩ − m − 1+ κ ∥ ϕ ∥ X > 0 . 22 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL Figure 5.1. The con tours Γ R , Γ R, ± , Γ R,ε , Γ R, high and Γ ε, low (in gr ey a r egion which c ontains the sp e ctrum of A ) . By con tinuit y , there exists a non-trivial interv al [ t 1 , t 2 ] around t 0 suc h that for all t ∈ [ t 1 , t 2 ], using (5.7) we ha v e lim inf R →∞  ∥ T Y e tA T X ϕ ∥ Y − ∥ T Y U R ( t ) T X ϕ ∥ Y  ≥ ∥ T Y e tA T X ϕ ∥ Y − C ⟨ t ⟩ − m − 1+ κ ∥ ϕ ∥ X ≥ η 2 . This gives a con tradiction with (5.6). • Splitting of c ontour inte gr al: By (5.1) there exist γ > 0 and C 1 > 0 such that for any λ ∈ C with Re λ ≥ − 2 γ and | Im( λ ) | ≥ τ 0 w e hav e λ ∈ ρ ( A ) and   ( A − λ ) − 1   B ( K ) ≤ C 1 . (5.8) W e consider θ ∈ C ∞ ( R ; [0 , 1]) supp orted in [ − 2 τ 0 , 2 τ 0 ] with θ = 1 on [ − τ 0 , τ 0 ]. Let ε ∈ (0 , 1]. F or τ ∈ R we set θ ε ( τ ) = − γ + ( ε + γ ) θ ( τ ) ∈ [ − γ , ε ] . In particular, θ ε ( τ ) = ε if | τ | ≤ τ 0 and θ ε ( τ ) = − γ if | τ | ≥ 2 τ 0 . Then for R ≥ 2 τ 0 w e consider the contour Γ R,ε deﬁned by the parametrization Γ R,ε :  [ − R, R ] → C τ 7→ θ ε ( τ ) + i τ . W e also consider the contours Γ R, − and Γ R, + deﬁned as the line segmen ts joining µ − i R to − γ − i R and − γ + i R to µ + i R , resp ectiv ely (see Figure 5.1). Then we ha ve U R ( t ) = U R, − ( t ) + U R,ε ( t ) + U R, + ( t ) , t ≥ 0 , where for ∗ ∈ {− , ε, + } we ha v e set U R, ∗ ( t ) = − 1 2i π Z Γ R, ∗ e tλ ( A − λ ) − 1 d λ. • Eliminating c ontribution of Γ R, ± in (5.7) : Let ϕ ∈ X and ψ ∈ Dom( A ). Given a ﬁxed λ 0 ∈ ρ ( A ) w e hav e, for λ ∈ ρ ( A ), ( A − λ ) − 1 ψ = 1 λ − λ 0  ( A − λ ) − 1 ( A − λ 0 ) − I  ψ . SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 23 With (5.8), this implies that there exists C 2 > 0 indep enden t of t ≥ 0, R > 0 and ψ ∈ Dom( A ) suc h that ∥ U R, ± ( t ) ψ ∥ K ≤ C 2 e tµ ( ∥ Aψ ∥ K + ∥ ψ ∥ K ) R . On the other hand, for some C 3 > 0 indep enden t of ϕ ∈ X , t ≥ 0 and R > 0, (5.8) giv es that ∥ U R, ± ( t )( ψ − T X ϕ ) ∥ K ≤ C 3 e tµ ∥ ψ − T X ϕ ∥ K , and so ∥ U R, ± ( t ) T X ϕ ∥ K ≤ C 2 e tµ ( ∥ Aψ ∥ K + ∥ ψ ∥ K ) R + C 3 e tµ ∥ ψ − T X ϕ ∥ K . Cho osing ﬁrst ψ ∈ Dom( A ) close to T X ϕ in K and then R large, w e see that for all t ≥ 0 and ϕ ∈ X w e hav e ∥ T Y U R, ± ( t ) T X ϕ ∥ Y − − − − → R →∞ 0 . (5.9) No w (5.7) will follow from the estimate lim sup R →∞ ∥ T Y U R,ε ( t ) T X ϕ ∥ Y ≤ C 4 e εt ⟨ t ⟩ − m − 1+ κ ∥ ϕ ∥ X , (5.10) where the constant C 4 > 0 is indep enden t of ϕ ∈ X , t ≥ 0 and ε > 0. Indeed, (5.9) and (5.10) imply that (5.10) holds with U R ( t ) instead of U R,ε ( t ) (the left-hand side not dep ending on ε ), so we can let ε → 0 in this estimate to arrive at (5.7). • Splitting of r elevant c ontour Γ R,ε : Deﬁne Γ ε, low as the restriction of Γ R,ε to τ ∈ [ − 2 τ 0 , 2 τ 0 ] (whic h do es not dep end on R ≥ 2 τ 0 ), and let Γ R, high b e the restriction to τ ∈ [ − R, R ] \ [ − 2 τ 0 , 2 τ 0 ] (which does not dep end on ε ). Let U R, high ( t ) = − 1 2i π Z Γ R, high e tλ ( A − λ ) − 1 d λ = − e − γ t 2 π Z 2 τ 0 ≤| τ |≤ R e i tτ ( A − ( − γ + i τ )) − 1 d τ and U ε, low ( t ) = − 1 2i π Z Γ ε, low e tλ ( A − λ ) − 1 d λ = − 1 2i π Z | τ |≤ 2 τ 0 e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i)( A − ( θ ε ( τ ) + i τ )) − 1 d τ . Then we ha v e U R,ε ( t ) = U ε, low ( t ) + U R, high ( t ) , t ≥ 0 . • (5.10) for U R, high : W e b egin with the con tribution of high frequencies. By (5.5) and Plancherel theorem (see [49, Thm. 5.1.4]), we ha v e for ϕ ∈ X Z R ∥ ( A − ( µ + i τ )) − 1 T X ϕ ∥ 2 K d τ ≤ M 2 2 π Z ∞ 0 e − 2 t ( µ − ω ) ∥ T X ϕ ∥ 2 K d t = M 2 ∥ T X ϕ ∥ 2 K 4 π ( µ − ω ) . F or | τ | ≥ 2 τ 0 w e hav e ( A − ( − γ + i τ )) − 1 =  1 − ( µ + γ )( A − ( − γ + i τ )) − 1  ( A − ( µ + i τ )) − 1 , so, by (5.8), ∥ ( A − ( − γ + i τ )) − 1 T X ϕ ∥ K ≤  1 + C 1 ( µ + γ )  ∥ ( A − ( µ + i τ )) − 1 T X ϕ ∥ K . After integration w e get Z | τ |≥ 2 τ 0 ∥ ( A − ( − γ + i τ )) − 1 T X ϕ ∥ 2 K d τ ≤ M 2  1 + C 1 ( µ + γ )  2 4 π ( µ − ω ) ∥ T X ϕ ∥ 2 K . (5.11) 24 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL The adjoint A ∗ of A is the generator of the semigroup  (e tA ) ∗  t ≥ 0 (see [20, Thm. 7.3.3]), whic h has the same growth b ound as (e tA ) t ≥ 0 . Moreov er, for all | τ | ≥ 2 τ 0 , w e hav e b y (5.8) ∥ ( A ∗ − ( − γ − i τ )) − 1 ∥ B ( K ) = ∥ ( A − ( − γ + i τ )) − 1 ∥ B ( K ) ≤ C 1 , so for ψ ∈ Y ∗ w e similarly get Z | τ |≥ 2 τ 0 ∥ ( A ∗ − ( − γ − i τ )) − 1 T ∗ Y ψ ∥ 2 K d τ ≤ M 2  1 + C 1 ( µ + γ )  2 4 π ( µ − ω ) ∥ T ∗ Y ψ ∥ 2 K , (5.12) where we use the iden tiﬁcation K ∗ ≃ K . Now for ϕ ∈ X and ψ ∈ Y ∗ w e hav e ( − i t ) ⟨ U R, high ( t ) T X ϕ, T ∗ Y ψ ⟩ K = − e − γ t 2 π Z 2 τ 0 ≤| τ |≤ R  − ∂ τ e i tτ  ⟨ ( A − ( − γ + i τ )) − 1 T X ϕ, T ∗ Y ψ ⟩ K d τ . W e integrate by parts. The b oundary terms can b e neglected since they are exp o- nen tially decaying in t . More precisely , for | τ | ∈ { 2 τ 0 , R } we ha v e by (5.8)     e − γ t 2 π e i tτ ⟨ ( A − ( − γ + i τ )) − 1 T X ϕ, T ∗ Y ψ ⟩ K     ≤ e − γ t 2 π C 1 ∥ T X ϕ ∥ K ∥ T ∗ Y ψ ∥ K . Then, by the Cauc h y-Sc h warz inequalit y and (5.11)–(5.12),      e − γ t 2 π Z 2 τ 0 ≤| τ |≤ R e i tτ ∂ τ ⟨ ( A − ( − γ + i τ )) − 1 T X ϕ, T ∗ Y ψ ⟩ K d τ      ≤ e − γ t 2 π Z 2 τ 0 ≤| τ |≤ R   ⟨ ( A − ( − γ + i τ )) − 2 T X ϕ, T ∗ Y ψ ⟩ K   d τ ≤ e − γ t 2 π Z | τ |≥ 2 τ 0 ∥ ( A − ( − γ + i τ )) − 1 T X ϕ ∥ 2 d τ ! 1 2 × Z | τ |≥ 2 τ 0 ∥ ( A ∗ − ( − γ − i τ )) − 1 T ∗ Y ψ ∥ 2 d τ ! 1 2 ≤ C 5 e − γ t ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ , for a constant C 5 > 0 indep endent of t , R , ϕ and ψ . Finally , b y the ab o ve there exists C 6 > 0 suc h that for all t ≥ 0, ϕ ∈ X and R > 0 we ha v e ∥ T Y U R, high ( t ) T X ϕ ∥ Y = sup 0  = ψ ∈ Y ∗ |  ψ , T Y U R, high ( t ) T X ϕ  Y ∗ × Y | ∥ ψ ∥ Y ∗ ≤ C 6 e − tγ ∥ ϕ ∥ X . (5.13) • (5.10) for U ε, low : Now we estimate the contribution of low frequencies. It is enough to sho w (5.10) for t ≥ 1. Notice ﬁrst that due to (5.1), (5.8) and | Γ ε, low ( τ ) | ≥ | τ | , (5.2) is v alid (with a possibly diﬀerent constan t) for λ running along the curve Γ ε, low and | λ | replaced by | τ | on the right-hand side. SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 25 Let ϕ ∈ X and ψ ∈ Y ∗ b e arbitrary . In tegrating by parts, for j ∈ N 0 , we start b y deriving Z | τ |≤ 2 τ 0 e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − j T X ϕ, T ∗ Y ψ ⟩ K d τ = 1 t Z | τ |≤ 2 τ 0 ∂ τ  e t ( θ ε ( τ )+i τ )  ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − j T X ϕ, T ∗ Y ψ ⟩ K d τ = j + 1 ( − t ) Z | τ |≤ 2 τ 0 e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − ( j +1) T X ϕ, T ∗ Y ψ ⟩ K d τ − 1 ( − t ) h e t ( θ ε ( τ )+i τ ) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − j T X ϕ, T ∗ Y ψ ⟩ K i 2 τ 0 − 2 τ 0 . By an inductive argumen t, for m ∈ N 0 w e get − 2i π ⟨ U ε, low ( t ) T X ϕ, T ∗ Y ψ ⟩ K = Z | τ |≤ 2 τ 0 e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 T X ϕ, T ∗ Y ψ ⟩ K d τ = m ! ( − t ) m Z | τ |≤ 2 τ 0 e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − m T X ϕ, T ∗ Y ψ ⟩ K d τ − m − 1 X j =0 j ! ( − t ) j +1 h e t ( θ ε ( τ )+i τ ) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − j T X ϕ, T ∗ Y ψ ⟩ K i 2 τ 0 − 2 τ 0 . Note that by (5.8) and since θ ε ( ± 2 τ 0 ) = − γ , the b oundary terms deca y exp onen- tially in time and can th us b e neglected. It remains to estimate the integral on the righ t hand side, where we split the in tegration at | τ | = 1 /t ≤ 1 ≤ τ 0 . F or the integral around zero, w e use (5.2) with j = m to obtain      Z | τ |≤ 1 t e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − m T X ϕ, T ∗ Y ψ ⟩ K d τ      ≲ e εt Z | τ |≤ 1 t | τ | − κ d τ ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ ≲ e εt t κ − 1 ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ ; recall that θ ′ ε ( τ ) = ( ε + γ ) θ ′ ( τ ) is uniformly b ounded in ε . T o estimate the remaining in tegral, w e in tegrate b y parts once more and obtain Z 1 t ≤| τ |≤ 2 τ 0 e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − m T X ϕ, T ∗ Y ψ ⟩ K d τ = m + 1 − t Z 1 t ≤| τ |≤ 2 τ 0 e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − ( m +1) T X ϕ, T ∗ Y ψ ⟩ K d τ + 1 t h e t ( θ ε ( τ )+i τ ) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − m T X ϕ, T ∗ Y ψ ⟩ K i 2 τ 0 − 2 τ 0 − 1 t h e t ( θ ε ( τ )+i τ ) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − m T X ϕ, T ∗ Y ψ ⟩ K i 1 t − 1 t . As before, the b oundary terms at τ = ± 2 τ 0 are deca ying exponentially in time. Using that θ ε ( ± 1 /t ) = ε together with (5.2) for j = m , the b oundary terms at τ = ± 1 /t are dominated by e εt t κ − 1 ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ . 26 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL Altogether, in the case κ  = 0, it follo ws from (5.2) with j = m + 1 that      Z 1 t ≤| τ |≤ 2 τ 0 e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − m T X ϕ, T ∗ Y ψ ⟩ K d τ      ≲ e εt t κ − 1 ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ + e εt t − 1 Z 1 t ≤| τ |≤ 2 τ 0 | τ | − κ − 1 d τ ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ ≲ e εt t κ − 1 ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ . If κ = 0 w e integrate by parts once more (notice that the b ound (5.2) for j = m +2 is only used in this case). As b efore, the boundary terms at ± 2 τ 0 are exp onentially deca ying, while from (5.2) with j = m + 1 w e see that the b oundary terms at ± 1 /t are dominated by e εt t − 1 ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ , so we can write Z 1 t ≤| τ |≤ 2 τ 0 e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − m T X ϕ, T ∗ Y ψ ⟩ K d τ = ( m + 1)( m + 2) t 2 Z 1 t ≤| τ |≤ 2 τ 0 e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i) × ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − ( m +2) T X ϕ, T ∗ Y ψ ⟩ K d τ + O (e εt t − 1 ) ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ . As ab o v e, we get from (5.2) with j = m + 2 that      Z 1 t ≤| τ |≤ 2 τ 0 e t ( θ ε ( τ )+i τ ) ( θ ′ ε ( τ ) + i) ⟨ ( A − ( θ ε ( τ ) + i τ )) − 1 − m T X ϕ, T ∗ Y ψ ⟩ K d τ      ≲ e εt t − 1 ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ + e εt t − 2 Z 1 t ≤| τ |≤ 2 τ 0 | τ | − 2 d τ ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ ≲ e εt t − 1 ∥ ϕ ∥ X ∥ ψ ∥ Y ∗ . In any case, as in (5.13) for high frequencies, it follows that there exists C 7 > 0 suc h that for all t ≥ 0, ε > 0 and ϕ ∈ X w e hav e ∥ T Y U ε, low ( t ) T X ϕ ∥ Y ≤ C 7 e εt ⟨ t ⟩ − m − 1+ κ ∥ ϕ ∥ X . (5.14) With (5.13) and (5.14) we deduce (5.10), which concludes the pro of. □ 6. The case of const ant damping and optimality of the deca y In this section w e prov e optimality for the estimates of Theorem 1.2 by consid- ering the mo del case q ( x ) ≡ 0 and a ( x ) ≡ 1 on the full Euclidean space Ω = R d . More precisely , w e pro ve the following result (see also Remark 6.4 below), where for j ∈ { 0 , 1 , 2 } we ha v e set c j =  sup s ≥ 0 s j +1 e − 2 s  1 2 =  j + 1 2e  j +1 2 . Prop osition 6.1. Supp ose Ω = R d , q ( x ) ≡ 0 and a ( x ) ≡ 1 . L et ρ 0 , ρ 1 , ρ 2 : R + → R + b e such that for al l F ∈ K and al l t > 0 we have ∥ u ( t ) ∥ L 2 ≤ ρ 0 ( t ) ∥ F ∥ K , ∥∇ u ( t ) ∥ L 2 ≤ ρ 1 ( t ) ∥ F ∥ K , ∥ ∂ t u ( t ) ∥ L 2 ≤ ρ 2 ( t ) ∥ F ∥ K , (6.1) wher e K is as in (1.7) – (1.8) and u ( t ) is the solution of (1.1) . F or every ε > 0 ther e exists t 0 ≥ 1 such that for al l t ≥ t 0 we have ρ 0 ( t ) ≥ c 0 − ε √ t , ρ 1 ( t ) ≥ c 1 − ε t , ρ 2 ( t ) ≥ c 2 − ε t 3 2 . SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 27 T o prov e Prop osition 6.1, we ﬁrst chec k that u ( t ) behav es for large times like a solution of the heat equation. Lemma 6.2. Supp ose Ω = R d , q ( x ) ≡ 0 and a ( x ) ≡ 1 . L et F = ( f , g ) ∈ K with K as in (1.7) – (1.8) , and let u ( t ) b e the solution of (1.1) . F or t ≥ 0 we set u 0 ( t ) = e t ∆ ( f + g ) . Then ther e exists c > 0 such that for al l t ≥ 1 we have ∥ u ( t ) − u 0 ( t ) ∥ L 2 ≤ ct − 1 ∥ F ∥ H 1 ⊕ L 2 , ∥∇ ( u ( t ) − u 0 ( t )) ∥ L 2 ≤ ct − 3 2 ∥ F ∥ H 1 ⊕ L 2 , ∥ ∂ t ( u ( t ) − u 0 ( t )) ∥ L 2 ≤ ct − 2 ∥ F ∥ H 1 ⊕ L 2 . (6.2) Pr o of. F or t ≥ 0 w e recall that U ( t ) = e t A F = ( u ( t ) , ∂ t u ( t )). W e prov e (6.2) for the corresp onding F ourier transformed quan tities (with respect to the space v ariable x ), uniformly in F ∈ K and t ≥ 1 (ﬁxed but arbitrary). F or ξ ∈ R d w e hav e b U ( t, ξ ) = e tM ( ξ ) b F ( ξ ) , M ( ξ ) =  0 1 − | ξ | 2 − 1  , as well as b u 0 ( t, ξ ) = e − t | ξ | 2 ( b f ( ξ ) + b g ( ξ )) . (6.3) Moreo ver, w e ha ve d ∂ t u 0 ( t, ξ ) = ∂ t b u 0 ( t, ξ ) = −| ξ | 2 b u 0 ( t, ξ ) . (6.4) Corresp onding to the w eigh t in the norm in F ( H 1 ( R d ) ⊕ L 2 ( R d )), for ξ ∈ R d and ζ = ( ζ 1 , ζ 2 ) ∈ C 2 w e set | ζ | 2 C 2 ξ := ( | ξ | 2 + 1) | ζ 1 | 2 + | ζ 2 | 2 . T o prov e the F ourier analogues of (6.2), we split the in tegration in three regions: (i) high frequencies | ξ | > 1; (ii) in termediate frequencies δ ≤ | ξ | ≤ 1, with δ ∈ (0 , 1 / 8) selected suitably in (6.16); (iii) small frequencies | ξ | < δ . The main con tribution comes from the part in (iii), while (i) and (ii) lead to exp o- nen tially small terms in time (b oth for b u ( t ) and b u 0 ( t ) separately). Note that in (ii) and (iii), the Euclidean norm | · | C 2 is equiv alen t to | · | C 2 ξ (with uniform constants in ξ ). • Basic estimates and exp ansions for b U ( t ) : The eigenv alues of M ( ξ ) read λ ± ( ξ ) = − 1 ± q 1 − 4 | ξ | 2 2 , λ + ( ξ ) − λ − ( ξ ) = p 1 − 4 | ξ | 2 , (with one double eigenv alue if 4 | ξ | 2 = 1). Their real parts satisfy Re λ − ( ξ ) ≤ − 1 2 , ξ ∈ R d , Re λ ± ( ξ ) = − 1 2 , | ξ | ≥ 1 4 , Re λ + ( ξ ) ≤ − 1 + √ 1 − 4 δ 2 2 := − η ≡ − η ( δ ) < 0 , | ξ | ≥ δ. (6.5) When 4 | ξ | 2  = 1, we can write e tM ( ξ ) = e tλ + ( ξ )  Ψ + ( ξ ) , ·  C 2 Φ + ( ξ ) + e tλ − ( ξ )  Ψ − ( ξ ) , ·  C 2 Φ − ( ξ ) , (6.6) where Φ ± ( ξ ) =  1 λ ± ( ξ )  , Ψ ± ( ξ ) = ± 1 λ + ( ξ ) − λ − ( ξ )  − λ ∓ ( ξ ) 1  . 28 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL W e hav e | λ + ( ξ ) − λ − ( ξ ) | ≳ | ξ | , | λ ± ( ξ ) | ≤ | Φ ± ( ξ ) | C 2 ξ ≲ | ξ | , | ξ | > 1 , (6.7) as well as | λ + ( ξ ) − λ − ( ξ ) | ≳ 1 , | Ψ − ( ξ ) | C 2 | Φ − ( ξ ) | C 2 ≲ 1 , | ξ | < δ , (6.8) (indep enden tly of δ < 1 / 8). One moreov er computes λ + ( ξ ) = − | ξ | 2 + O ( | ξ | 4 ) , ξ → 0 , (6.9) and ﬁnally Ψ + ( ξ ) =  1 1  + O ( | ξ | 2 ) , Φ + ( ξ ) =  1 − | ξ | 2 + O ( | ξ | 4 )  , ξ → 0 . (6.10) • (i) High / (ii) interme diate fr e quencies for b u 0 ( t ) : F or every | ξ | ≥ δ , using (6.3) w e estimate | ξ b u 0 ( t, ξ ) | ≤ 1 √ t p t | ξ | 2 e − t 2 | ξ | 2 e − t 2 | ξ | 2 | b F ( ξ ) | C 2 ξ ≲ e − δ 2 2 t | b F ( ξ ) | C 2 ξ . (6.11) Similarly , considering (6.4), one derives | d ∂ t u 0 ( t, ξ ) | ≲ e − δ 2 2 t | b F ( ξ ) | C 2 ξ , | ξ | ≥ δ. (6.12) By integration, it follo ws from (6.3), (6.11) and (6.12) that ∥ b u 0 ( t ) ∥ L 2 ( | ξ |≥ δ ) + ∥ ξ b u 0 ( t ) ∥ L 2 ( | ξ |≥ δ ) + ∥ d ∂ t u 0 ( t ) ∥ L 2 ( | ξ |≥ δ ) ≲ e − δ 2 2 t ∥| b F ( ξ ) | C 2 ξ ∥ L 2 ( | ξ |≥ δ ) ≤ e − δ 2 2 t ∥ F ∥ H 1 ⊕ L 2 . (6.13) • (i) High fr e quencies for b U ( t ) : F rom (6.7), w e obtain for | ξ | > 1 that   ⟨ Ψ ± ( ξ ) , b F ( ξ ) ⟩ C 2   ≲ 1 | λ + ( ξ ) − λ − ( ξ ) |  | λ ∓ ( ξ ) || b f ( ξ ) | + | b g ( ξ ) |  ≲ | b f ( ξ ) | + | ξ | − 1 | b g ( ξ ) | ≲ | ξ | − 1 | b F ( ξ ) | C 2 ξ . Com bining this with (6.6), (6.5) and (6.7), we get | b U ( t ) | C 2 ξ = | e tM ( ξ ) b F ( ξ ) | C 2 ξ ≲ e − t 2 | b F ( ξ ) | C 2 ξ , | ξ | > 1 . In tegrating this estimate leads to ∥ b u ( t ) ∥ L 2 ( | ξ | > 1) + ∥ ξ b u ( t ) ∥ L 2 ( | ξ | > 1) + ∥ d ∂ t u ( t ) ∥ L 2 ( | ξ | > 1) ≲ e − t 2 ∥ F ∥ H 1 ⊕ L 2 . (6.14) • (ii) Interme diate fr e quencies for b U ( t ) : F or every ﬁxed ξ 0 ∈ R d suc h that δ ≤ | ξ 0 | ≤ 1, b y computing the matrix exp onen tial of M ( ξ 0 ) and considering (6.5), there exists C ξ 0 > 0 suc h that ∥ e tM ( ξ 0 ) ∥ B ( C 2 ) ≤ C ξ 0 e − η t (1 + t ) ≲ C ξ 0 e − η 2 t ; note that the factor (1 + t ) is due to the Jordan blo ck structure for | ξ 0 | = 1 / 2. By con tinuit y of M , there exists a neigh borho o d V ξ 0 of ξ 0 suc h that this estimate holds for all ξ ∈ V ξ 0 with η / 2 replaced by η / 4. Finally , by compactness, we obtain ∥ e tM ( ξ ) ∥ B ( C 2 ) ≲ e − η 4 t , δ ≤ | ξ | ≤ 1 . Similarly as in (i), up on suitable in tegration (and using the uniform equiv alence b et w een | · | C 2 and | · | C 2 ξ ) this leads to ∥ b u ( t ) ∥ L 2 ( δ ≤| ξ |≤ 1) + ∥ ξ b u ( t ) ∥ L 2 ( δ ≤| ξ |≤ 1) + ∥ d ∂ t u ( t ) ∥ L 2 ( δ ≤| ξ |≤ 1) ≲ e − η 4 t ∥ F ∥ H 1 ⊕ L 2 . (6.15) SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 29 • (iii) L ow fr e quencies for b u ( t ) − b u 0 ( t ) : Considering (6.9), w e can select δ suﬃ- cien tly small such that | λ + ( ξ ) + | ξ | 2 | ≤ | ξ | 2 2 , | ξ | < δ ; (6.16) note also that λ + ( ξ ) ≤ 0. It follows that, for | ξ | < δ ,   e tλ + ( ξ ) − e − t | ξ | 2   =   λ + ( ξ ) + | ξ | 2   Z t 0 e − t | ξ | 2  1 − s t λ + ( ξ )+ | ξ | 2 | ξ | 2  d s (6.17) ≤   λ + ( ξ ) + | ξ | 2   t e − t 2 | ξ | 2 = t e − t 2 | ξ | 2 O ( | ξ | 4 ) , ξ → 0 . Using (6.10) and (6.3), for | ξ | < δ , w e estimate e tλ + ( ξ )  Ψ + ( ξ ) , b F ( ξ )  C 2 =  e tλ + ( ξ ) − e − t | ξ | 2   Ψ + ( ξ ) , b F ( ξ )  C 2 + e − t | ξ | 2  Ψ + ( ξ ) , b F ( ξ )  C 2 =  e tλ + ( ξ ) − e − t | ξ | 2   b f ( ξ ) + b g ( ξ ) + O  | ξ | 2 | b F ( ξ ) | C 2   + b u 0 ( t, ξ ) + e − t | ξ | 2 O  | ξ | 2 | b F ( ξ ) | C 2  , ξ → 0 . (6.18) F rom the ﬁrst comp onen t of (6.6), by combining (6.18), (6.17), (6.5), (6.8) and using | · | C 2 ≈ | · | C 2 ξ , we compute   b u ( t, ξ ) − b u 0 ( t, ξ )   ≤    e tλ + ( ξ )  Ψ + ( ξ ) , b F ( ξ )  C 2 − b u 0 ( t, ξ )    + | e tλ − ( ξ ) || Ψ − ( ξ ) | C 2 | Φ − ( ξ ) | C 2 | b F ( ξ ) | C 2 ≲  e − t | ξ | 2 O ( | ξ | 2 ) + t e − t 2 | ξ | 2 O ( | ξ | 4 ) + e − t 2  | b F ( ξ ) | C 2 ξ , ≲ 1 t  t | ξ | 2 e − t | ξ | 2 + ( t | ξ | 2 ) 2 e − t 2 | ξ | 2 + t e − t 2  | b F ( ξ ) | C 2 ξ , (6.19) for all | ξ | < δ . After integration, and using that the last brac ket on the righ t abov e is uniformly b ounded in ξ and t , we ha v e ∥ b u ( t ) − b u 0 ( t ) ∥ L 2 ( | ξ | <δ ) ≲ 1 t ∥ F ∥ H 1 ⊕ L 2 . (6.20) By a similar argument, (6.19) implies that   ξ  b u ( t ) − b u 0 ( t )    L 2 ( | ξ | <δ ) ≲ 1 t 3 2 ∥ F ∥ H 1 ⊕ L 2 . (6.21) F or the remaining estimate, using the second comp onen t of (6.6) and combining (6.10), (6.4), (6.18), (6.17), (6.5), (6.8) and | · | C 2 ≈ | · | C 2 ξ , we obtain   d ∂ t u ( t, ξ ) − d ∂ t u 0 ( t, ξ )   ≲   e tλ + ( ξ )  Ψ + ( ξ ) , b F ( ξ )  C 2  − | ξ | 2 + O ( | ξ | 4 )  + | ξ | 2 b u 0 ( t, ξ )   + | e tλ − ( ξ ) || Ψ − ( ξ ) | C 2 | Φ − ( ξ ) | C 2 | b F ( ξ ) | C 2 ≲  e − t | ξ | 2 O ( | ξ | 4 ) + t e − t 2 | ξ | 2 O ( | ξ | 6 ) + e − t 2  | b F ( ξ ) | C 2 ξ ≲ 1 t 2  ( t | ξ | 2 ) 2 e − t | ξ | 2 + ( t | ξ | 2 ) 3 e − t 2 | ξ | 2 + t 2 e − t 2  | b F ( ξ ) | C 2 ξ , for all | ξ | < δ . Hence, after in tegration, ∥ d ∂ t u ( t ) − d ∂ t u 0 ( t ) ∥ L 2 ( | ξ | <δ ) ≲ 1 t 2 ∥ F ∥ H 1 ⊕ L 2 . (6.22) • Conclusion: The proof is completed b y com bining (6.13), (6.14), (6.15), (6.20), (6.21), (6.22) and unitarity of the F ourier transform. □ 30 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL W e deﬁne K = H 1 ( R d ) ∩ ˙ H − 1 ( R d ), with norm ∥ f ∥ 2 K = ∥ f ∥ 2 H 1 ( R d ) + ∥ f ∥ 2 ˙ H − 1 ( R d ) = Z R d  | ξ | 2 + 1 + | ξ | − 2    b f ( ξ )   2 d ξ . Lemma 6.3. Supp ose Ω = R d , q ( x ) ≡ 0 and a ( x ) ≡ 1 . L et e ρ 0 , e ρ 1 , e ρ 2 : R + → R + and T 0 > 0 b e such that for al l f 0 ∈ K and t ≥ T 0 we have   e t ∆ f 0   L 2 ≤ e ρ 0 ( t ) ∥ f 0 ∥ K ,   ∇ e t ∆ f 0   L 2 ≤ e ρ 1 ( t ) ∥ f 0 ∥ K ,   ∂ t e t ∆ f 0   L 2 =   ∆e t ∆ f 0   L 2 ≤ e ρ 2 ( t ) ∥ f 0 ∥ K . F or every ε > 0 ther e exists t 0 ≥ T 0 such that for al l j ∈ { 0 , 1 , 2 } and t ≥ t 0 we have e ρ j ( t ) ≥ c j − ε t j +1 2 . (6.23) Pr o of. Let j ∈ { 0 , 1 , 2 } and, without loss of generalit y , consider ε ∈  0 , c j 2  with ε ≤ 4 T 0 . Let 0 ≤ η 1 < η 2 ≤ 2 b e suc h that inf s ∈ [ η 1 ,η 2 ] s j +1 e − 2 s ≥  c j − ε 2  2 , (notice that the maxim um v alue c 2 j ∈ (0 , 1) is reached at j +1 2 ∈ (0 , 2)). F or ﬁxed t ≥ t 0 := 4 ε ≥ T 0 , we consider f 0 ∈ S ( R d ) such that b f 0 ( ξ )  = 0 = ⇒ η 1 ≤ t | ξ | 2 ≤ η 2 . In particular, we ha v e f 0 ∈ K . Notice that if | ξ | 2 ≤ η 2 t ≤ ε 2 , then  c j − ε 2  2 | ξ | 2 − ( c j − ε ) 2 1 | ξ | 2 + 1 + | ξ | 2 ! ≥ ε ( c j − ε ) | ξ | 2 1 − ( c j − ε ) | ξ | 2 + | ξ | 4 ε ! ≥ 0; recall that c j ∈ (0 , 1) and ε < c j 2 . Then for all ξ ∈ R d w e hav e | ξ | 2 j e − 2 t | ξ | 2 | b f 0 ( ξ ) | 2 = e − 2 t | ξ | 2 ( t | ξ | 2 ) j +1 1 [ η 1 ,η 2 ] ( t | ξ | 2 ) | b f 0 ( ξ ) | 2 t j +1 | ξ | 2 ≥  c j − ε 2  2 | b f 0 ( ξ ) | 2 t j +1 | ξ | 2 1 [ η 1 ,η 2 ] ( t | ξ | 2 ) ≥ ( c j − ε ) 2 t j +1 1 | ξ | 2 + 1 + | ξ | 2 ! | b f 0 ( ξ ) | 2 . After integration o v er ξ ∈ R d and by the Planc herel theorem, e ρ j ( t ) ∥ f 0 ∥ K ≥    ( − ∆) j 2 e t ∆ f 0    L 2 ≥ c j − ε t j +1 2 ∥ f 0 ∥ K . □ Finally we can pro v e Prop osition 6.1. Pr o of of Pr op osition 6.1. Let ρ 0 : R + → R + suc h that (6.1) holds and let f ∈ K ; notice that then F = ( f , 0) ∈ K and ∥ F ∥ K = ∥ f ∥ K . Let c > 0 b e giv en b y Lemma 6.2 and let T 0 ≥ 1 suc h that ct − 1 ≤ ε 2 √ t , t ≥ T 0 . Then for all t ≥ T 0 w e hav e   e t ∆ f   L 2 = ∥ u 0 ( t ) ∥ L 2 ≤ ρ 0 ( t ) ∥ F ∥ K + ct − 1 ∥ F ∥ H 1 ⊕ L 2 ≤  ρ 0 ( t ) + ε 2 √ t  ∥ f ∥ K . SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 31 By Lemma 6.3 applied with ε 2 , there exists t 0 ≥ T 0 suc h that for all t ≥ t 0 ρ 0 ( t ) + ε 2 √ t ≥ c 0 − ε 2 √ t . The pro ofs for ρ 1 ( t ) and ρ 2 ( t ) are similar. □ Remark 6.4. F or every ε > 0, one can construct an initial condition which up to a factor t ε ac hieves the deca y rates in (6.23). T o see this, let χ ∈ C ∞ c ( R d ; [0 , 1]) with χ = 1 on B (0 , δ ) for some δ > 0. Let f 0 b e the inv erse F ourier transform of ξ 7→ | ξ | 1 − d 2 + ε χ ( ξ ). Then f 0 ∈ K and for t > 0 we ha v e   e t ∆ f 0   2 L 2 ≥ Z B (0 ,δ ) e − 2 t | ξ | 2 | ξ | 2 − d +2 ε d ξ = (2 t ) − (1+ ε ) Z B (0 , √ 2 tδ ) e −| η | 2 | η | 2 − d +2 ε d η , so, for all t ≥ t 0 with a ﬁxed t 0 > 0,   e t ∆ f 0   L 2 ≳ t − 1 2 − ε 2 . Similarly , for all t ≥ t 0 ,   ∇ e t ∆ f 0   L 2 ≳ t − 1 − ε 2 ,   ∂ t e t ∆ f 0   L 2 ≳ t − 3 2 − ε 2 . 7. Remarks on non-uniforml y positive damping W e hav e stated Proposition 4.1 ab out the high frequency estimate and hence the energy estimates of Theorem 1.2 under the assumption that the damping co eﬃcien t a is uniformly p ositive on Ω. In this section, we discuss some settings where this assumption is not satisﬁed. 7.1. Cases with the geometric con trol condition. If the (unbounded) damp- ing a satisﬁes the geometric con trol condition, it is expected that the resolven t norm of A is b ounded at ± i ∞ as in Prop osition 4.1. In the one-dimensional case, this was established in [5]. (The sp ectral prop ert y σ ( A ) ∩ i R ⊂ { 0 } , which implies (7.1) and is not discussed in [5], follows b y a simple ODE argument.) Theorem 7.1 ([5, Thm. 3.5]) . L et Ω = R and let A b e as in (2.2) and (2.7) . L et 0 ≤ a, q ∈ C ∞ ( R ) satisfy the fol lowing c onditions. (i) a is unb ounde d at inﬁnity: lim x →±∞ a ( x ) = + ∞ , (ii) a has c ontr ol le d derivatives: ∀ n ∈ N , ∃ C n > 0 , ∀ x ∈ R : | a ( n ) ( x ) | ≤ C n (1 + a ( x )) ⟨ x ⟩ − n , (iii) q has c ontr ol le d derivatives: ∀ n ∈ N , ∃ C ′ n > 0 , ∀ x ∈ R : | q ( n ) ( x ) | ≤ C ′ n (1 + q ( x )) ⟨ x ⟩ − n , (iv) q is eventual ly not bigger than a : ∃ x 0 ≥ 0 , ∃ K > 0 , ∀ | x | > x 0 : q ( x ) ≤ K a ( x ) . Then ∥ ( A − i b ) − 1 ∥ B ( H ) ≈ 1 , | b | → ∞ , and, for any ε > 0 , sup λ ∈ C + \ ε D ∥ ( A − λ ) − 1 ∥ B ( H ) < ∞ . (7.1) In higher dimensions with Ω = R d , the analogous uniform resolv en t bound for the damping a ( x ) = | x | β , β > 0, follo ws from a result [37, Thm.2.1] for Schr¨ odinger op erators with imaginary p oten tials − ∆ + i | x | β in L 2 ( R d ). 32 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL Prop osition 7.2. L et A b e as in (2.2) and (2.7) with Ω = R d , a ( x ) = | x | β , β > 0 , and q = 0 . Then, for any ε > 0 , sup λ ∈ C + \ ε D ∥ ( A − λ ) − 1 ∥ B ( H ) < ∞ . R emarks on the pr o of of Pr op osition 7.2. In [37, Thm. 2.1] it is prov ed in particular that ∥ ( − ∆ + i | x | β − µ ) − 1 ∥ B ( L 2 ) ≲ µ − β 2( β +1) , µ → + ∞ . (7.2) Consider only b > 0. Rescaling T i b = − ∆ + i b | x | β − b 2 with a parameter σ > 0 leads to ∥ T − 1 i b ∥ B ( L 2 ) = σ 2 ∥ ( − ∆ + i bσ 2+ β | x | β − σ 2 b 2 ) − 1 ∥ B ( L 2 ) . Selecting σ such that bσ 2+ β = 1, employing (7.2) and considering that T ∗ i b = T − i b (see [25, Thm. 2.4]), we arriv e at ∥ T − 1 − i b ∥ B ( L 2 ) = ∥ T − 1 i b ∥ B ( L 2 ) ≲ b − 1 , b → + ∞ . The claim then follows b y Lemma 4.2 as there are no purely imaginary eigenv al- ues (since T i b f = 0 implies b ∥ a 1 2 f ∥ 2 = Im t i b [ f ] = 0 and a ( x )  = 0 a.e. in R d , cf. Theorem 2.3 (i)). □ Since Assumption 3.1 is satisﬁed for b oth settings in Theorem 7.1 and Prop o- sition 7.2 (in the latter ev en Assumption 3.2 holds), the low frequency estimate from Theorem 3.3 applies and we arrive at the same conclusions for the deca y of solutions as in Theorem 1.2. Corollary 7.3. L et Ω , a and q b e as in The or em 7.1 or Pr op osition 7.2. Then the solution u ( t ) of (1.1) satisﬁes the de c ay estimates (1.9) – (1.11) in The or em 1.2, wher e in the setting of Pr op osition 7.2 even (1.12) – (1.13) hold. 7.2. Example without the geometric con trol condition. Non-uniformly pos- itiv e damping coeﬃcients which do not satisfy the geometric con trol condition ma y lead to a diﬀeren t sp ectral behavior for A at inﬁnity . W e discuss the following example on a tw o-dimensional strip, where the resolven t is no longer b ounded as λ = i b → ± i ∞ . In detail, let for some n ∈ N Ω = R × ( − 1 , 1) ⊂ R 2 , q ( x ) ≡ 0 , a ( x, y ) = x 2 n , ( x, y ) ∈ Ω . (7.3) Notice that a (0 , y ) ≡ 0, i.e. there is one tra jectory in Ω with no damping. This results in eigenv alues approac hing the imaginary axis at ± i ∞ with a rate dep end- ing on n (see Prop osition 7.4 b elow, and also [25, Prop. 6.3]). Consequen tly , the resolv ent of A is unbounded at ± i ∞ and w e show in Prop osition 7.5 that the rate predicted by the eigen v alues (7.7) is optimal. The pro ofs are based on the sp ectral equiv alence to the Sc h ur complement T λ , separation of v ariables and the subsequen t analysis of one-dimensional operators. In particular, for − ∂ 2 y sub ject to Dirichlet boundary conditions in L 2 ( − 1 , 1), we hav e an orthonormal basis of L 2 ( − 1 , 1) given by (normalized) eigenfunctions { g j } j ∈ N suc h that − g ′′ j = ζ j g j , ζ j =  j π 2  2 , g j ∈ H 1 0 ( − 1 , 1) ∩ H 2 ( − 1 , 1) , j ∈ N . (7.4) Hence, the Sch ur complemen t T λ = − ∆ + λx 2 n + λ 2 (7.5) SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 33 acting in L 2 (Ω) (deﬁned b y its quadratic form) is unitarily equiv alent to the or- thogonal sum of a sequence of op erators acting in L 2 ( R ), namely T λ ≃ M j ∈ N T λ,j , T λ,j := − ∂ 2 x + λx 2 n + λ 2 + ζ j , Dom( T λ,j ) := H 2 ( R ) ∩ Dom( x 2 n ) , j ∈ N , (7.6) (see Appendix B for details). This decomp osition allows us to characterize the sp ectrum of A as well as its resolv ent norm on the imaginary axis. Prop osition 7.4. L et A b e as in (2.2) and (2.7) with Ω , a and q as in (7.3) . Then σ e2 ( A ) = ( −∞ , 0] and, for every ﬁxe d k ∈ N 0 , ther e exists a se quenc e { λ k,j } j ∈ N with λ k,j , λ k,j ∈ σ p ( A ) having the asymptotic b ehavior λ k,j = i π 2 j + µ k 2  π j 2  − n n +1 e i π 2 ( n +2) ( n +1) + O k  j − 3 n +1 n +1  , j → ∞ . (7.7) Her e { µ k } k ∈ N 0 denote the eigenvalues of the self-adjoint anharmonic oscil lator − ∂ 2 x + x 2 n in L 2 ( R ) on the domain H 2 ( R ) ∩ Dom( x 2 n ) . Pr o of. Since (2.13) is satisﬁed, we conclude that σ ( A ) \ ( −∞ , 0] consists only of eigen v alues of ﬁnite m ultiplicities that may at most accumulate at ( −∞ , 0] and are symmetric with respect to the real axis. F urthermore, arguing as in [25, Prop. 6.3] where the case n = 1 was treated, w e hav e ( −∞ , 0] = σ e 2 ( A ). T o study the eigen v alues of A , we employ the sp ectral equiv alence (2.12) and the decomp osition (7.6), which in a straigh tforw ard wa y implies σ ( T λ ) = σ p ( T λ ) = [ j ∈ N σ p ( T λ,j ) . W e th us search for λ ∈ C with Re λ ≤ 0, Im λ > 0 and 0 ∈ σ p ( T λ,j ) for some j ∈ N . Employing a standard complex scaling argument for ﬁxed j ∈ N (see e.g. [5, Prop. 6.1] for details), one can prov e that 0 ∈ σ p ( T λ,j ) if and only if F ( λ ) ≡ F j,k ( λ ) := λ 2 + ζ j + µ k λ 1 n +1 = 0 (7.8) for some k ∈ N 0 and ζ j as in (7.4) . Here and in the following, the complex ( n + 1)-th ro ot is tak en as the holomorphic branc h ( · ) 1 n +1 : C \ ( −∞ , 0] → C , r e i φ 7→ r 1 n +1 e i φ n +1 , ϕ ∈ ( − π, π ) . (7.9) W e deﬁne F as a holomorphic function on the domain { z ∈ C : arg z ∈ ( π / 2 − ε, π ) } with some ε > 0, which con tains the range of sought solutions. W e ﬁx k ∈ N 0 . T o obtain the existence of the solutions to (7.8) and their asymptotic b eha vior as j → ∞ , we guess the ﬁrst terms of the expansion and apply the Rouch ´ e theorem suitably . F rom no w on, asymptotic relations are understo od as j → ∞ whic h will b e considered large enough if needed. Assume that λ is a solution of (7.8) for some large j . T aking p ow ers in (7.8), it follows that ( λ 2 + ζ j ) n +1 + ( − 1) n λµ n +1 k = 0 . Since | ζ j | ≈ j 2 , a solution λ is necessarily of order j and then, more precisely , | λ | ≈ j, r j := λ 2 + ζ j = O  j 1 n +1  . F rom this we further derive λ 2 = − ζ j + r j = − ζ j  1 + O  j − 2 n +1 n +1  , 34 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL whic h by the requiremen t Im λ > 0 then implies λ = i p ζ j + O  j − n n +1  . Using this relation, we ﬁnally arriv e at λ − i p ζ j = λ 2 + ζ j λ + i p ζ j = − µ k λ 1 n +1 2i p ζ j + O  j − n n +1  = − µ k 2  i p ζ j  1 n +1 i p ζ j  1 + O  j − 2 n +1 n +1   1 + O  j − 2 n +1 n +1  = − µ k 2  i p ζ j  − n n +1  1 + O  j − 2 n +1 n +1  . F rom the last iden tity w e can easily read oﬀ that the ﬁrst t wo terms in the expansion should b e λ 0 := i π 2 j + µ k 2  π j 2  − n n +1 e i π 2 n +2 n +1  = 0 . Considering a small parameter α ∈ C , we are looking for zeros of the function f ( α ) := F ( λ 0 + α ). Note that f is holomorphic for | α | ≤ 1 if j is large enough. A straigh tforward calculation yields f ( α ) = 2 αλ 0 + α 2 −  µ k 2  2  π j 2  − 2 n n +1 e i π 1 n +1 − µ k  π j 2  1 n +1 e i π 2 1 n +1 + µ k ( λ 0 + α ) 1 n +1 . Using the form ula ( z 1 z 2 ) 1 / ( n +1) = z 1 / ( n +1) 1 z 1 / ( n +1) 2 (whic h is justiﬁed with the con ven tion (7.9) when z 1 is purely imaginary and z 2 is in a small neighborho od of 1), we obtain ( λ 0 + α ) 1 n +1 =  i π j 2  1 n +1 (1 + x ) 1 n +1 , x = µ k 2  π j 2  − 2 n +1 n +1 e i π 2 1 n +1 − 2i α π j . The expansion (1 + x ) 1 n +1 = 1 + x n +1 + O ( | x | 2 ) then gives ( λ 0 + α ) 1 n +1 =  π j 2  1 n +1 e i π 2 1 n +1 + µ k 2( n + 1)  π j 2  − 2 n n +1 e i π 1 n +1 − i α n + 1  π j 2  − n n +1 e i π 2 1 n +1 + O k  j − 2 n +1 n +1  , where the remainder dep ends on k but is uniform in | α | ≤ 1. Putting together the ab o v e, one calculates f ( α ) = 2 αλ 0 + α 2 + µ k 2 e i π 1 n +1  1 n + 1 − µ k 2   π j 2  − 2 n n +1 − i µ k n + 1 e i π 2 1 n +1 α  π j 2  − n n +1 + O k  j − 2 n +1 n +1  . (7.10) W e apply the Rouch ´ e theorem (see e.g. [18, Thm. V.3.8]) to f and g : α 7→ 2 αλ 0 . T o this end, we b ound their diﬀerence by the sum of their mo duli on a circle | α | = C j − (3 n +1) / ( n +1) (matc hing the sought remainder) for some C ≡ C k > 0 to SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 35 b e selected suitably . F or such α , considering (7.10) we ha v e | f ( α ) − g ( α ) | = µ k 2     1 n + 1 − µ k 2      π j 2  − 2 n n +1  1 + O k ( j − 1 n +1 )  , | g ( α ) | = π C j − 2 n n +1  1 + O k ( j − 2 n +1 n +1 )  . (7.11) Cho osing ﬁrst C = µ k π     1 n + 1 − µ k 2      π 2  − 2 n n +1 , (dep ending only on k ), w e can ﬁnd j 0 suc h that, for every j ≥ j 0 w e hav e | α | ≤ 1 and (7.10) applies uniformly in α . It then follows from (7.11) that, p ossibly increasing j 0 , we ha v e | f ( α ) − g ( α ) | < | g ( α ) | , | α | = C j − 3 n +1 n +1 . By the Rouch ´ e theorem, w e conclude that, for ev ery j ≥ j 0 , f has exactly one zero α k,j with | α k,j | < C j − 3 n +1 n +1 , and in turn F has exactly one zero λ k,j with | λ k,j − λ 0 | ≤ C j − 3 n +1 n +1 , whic h is precisely the claim (7.7). □ Prop osition 7.5. L et A b e as in (2.2) and (2.7) with Ω , a and q as in (7.3) . Then ∥ ( A − i b ) − 1 ∥ B ( H ) ≲ | b | n n +1 , | b | → ∞ . (7.12) (Notic e that this r ate is optimal due to Pr op osition 7.4). Pr o of. W e show b elo w that ∥ T − 1 i b ∥ B ( L 2 (Ω)) ≲ | b | − 1 n +1 , b ∈ R \ { 0 } , and the claim then follows by Lemma 4.2. W e consider only b > 0, the case b < 0 follo ws b y using the fact that T ∗ λ = T λ (see [25, Thm. 2.4]). T o this end, we use that as a consequence of (7.6) (see App endix B for details), we ha v e ∥ T − 1 λ ∥ B ( L 2 (Ω)) = sup j ∈ N ∥ T − 1 λ,j ∥ B ( L 2 ( R )) . (7.13) Let j ∈ N b e ﬁxed. F or σ > 0 (to be c hosen suitably below) w e deﬁne the family of unitary op erators ( U σ u ) ( x ) := σ 1 2 u ( σ x ) , u ∈ L 2 ( R ) , x ∈ R . Then we ha v e  U σ T i b,j U − 1 σ u  ( x ) = − 1 σ 2 u ′′ ( x ) +  i bσ 2 n x 2 n − b 2 + ζ j  u ( x ) , x ∈ R , for any u ∈ U σ (Dom( T i b,j )). Setting σ := b − 1 2( n +1) , we deﬁne the operator family S b,j := σ 2 U σ T i b,j U − 1 σ = − ∂ 2 x + i x 2 n + σ 2 ( ζ j − b 2 ) . The op erator H n = − ∂ 2 x + i x 2 n , Dom( H n ) = H 2 ( R ) ∩ Dom  x 2 n  , is known to be m-accretiv e and it follo ws from [6, Eq. (7.2)] that there exist con- stan ts a 0 , K 0 > 0 suc h that ∥ ( H n − a ) − 1 ∥ B ( L 2 ( R )) ≤ K 0 a − n 2 n +1 , a ≥ a 0 . Moreo ver, Num( H n ) ⊂ { λ ∈ C : Re λ ≥ 0 , Im λ ≥ 0 } 36 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL and H n has no real eigenv alues (b y complex scaling, the eigen v alues of H n are complex-rotated eigenv alues of the self-adjoin t op erator − ∂ 2 x + x 2 n in L 2 ( R )). Hence, there exists K 1 > 0 suc h that ∥ ( H n − a ) − 1 ∥ B ( L 2 ( R )) ≤ K 1 , a ∈ R . F rom this, w e conclude that, for all j ∈ N and b > 0, also ∥ S − 1 b,j ∥ B ( L 2 ( R )) ≤ K 1 . Returning to T i b,j , we arriv e at ∥ T − 1 i b,j ∥ B ( L 2 ( R )) ≤ K 1 b − 1 n +1 , b > 0 , j ∈ N . □ T o deduce time decay from the previous resolv en t estimates, we employ the follo wing semigroup result allowing for the growth of the resolv ent norm at ± i ∞ . Theorem 7.6 ([9, Thm. 8.4]) . L et (e tA ) t ≥ 0 b e a b ounde d C 0 -semigr oup on a Hilb ert sp ac e H with gener ator A . Assume that σ ( A ) ∩ i R = { 0 } and that ther e exist α ≥ 1 and β > 0 such that ∥ ( A − i b ) − 1 ∥ B ( H ) = ( O ( | b | − α ) , | b | → 0 , O  | b | β  , | b | → ∞ . (7.14) Then ∥ e tA A α ( A − 1) − ( α + β ) ∥ B ( H ) = O  t − 1  , t → ∞ , and ∥ e tA A ( A − 1) − 2 ∥ B ( H ) = O  t − 1 γ  , t → ∞ , (7.15) wher e γ = max { α, β } . Conversely, if (7.15) holds for some γ > 0 , then (7.14) holds for α = max { γ , 1 } and β = γ . In conclusion, since the singularity of the resolven t at ± i ∞ in (7.12) is milder than at zero (see Theorem 3.3), the energy decay of the solutions for suitable initial data remains as for previous cases. Corollary 7.7. L et A b e as in (2.2) and (2.7) with Ω , a and q as in (7.3) . Then ther e exists C > 0 such that, for any initial data F = ( f , g ) ∈ e K := Dom( A ) ∩ Ran( A ) and t ≥ 0 , the solution u ( t ) of (1.1) satisﬁes ∥ ∂ t u ( t ) ∥ L 2 + ∥∇ u ( t ) ∥ L 2 ≤ C ∥ F ∥ e K ⟨ t ⟩ − 1 , wher e ∥ F ∥ 2 e K := ∥ F ∥ 2 H + ∥A F ∥ 2 H +   A − 1 F   2 H = ∥ F ∥ 2 K + ∥∇ g ∥ 2 L 2 + ∥ ∆ f − ag ∥ 2 L 2 . Pr o of. The claim follows by Theorem 7.6, in particular (7.15), with H = H and A = A . T o this end, note that Assumption 3.1 is satisﬁed, thus the resolven t estimate (3.5) holds. Moreov er, we ha v e also (7.12) and it is easy to see from (7.8) that σ ( A ) ∩ i R = { 0 } . Finally , similarly as in (5.4), we arriv e at ∥ e t A F ∥ H = ∥ e t A A ( A − 1) − 2 ( A − 1) 2 A − 1 F ∥ H ≲ ⟨ t ⟩ − 1 ∥A F − 2 F + A − 1 F ∥ H ≲ ⟨ t ⟩ − 1 ∥ F ∥ e K . □ 8. Comp arison with the resul t of Ikeha t a-T akeda As said in the in tro duction, Theorem 1.2 generalizes a previous result b y Ik ehata and T akeda, based on an approximation of the p ossibly un b ounded damping by a sequence of bounded dampings and a mo diﬁed Moraw etz m ultiplier metho d. Their precise result is the following. SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 37 Theorem 8.1 ([30, Thm. 1.2]) . L et Ω = R d with d ≥ 3 , let q = 0 and let a ∈ C ( R d ) with a ( x ) ≥ a 0 > 0 for al l x ∈ R d . If the initial data ( f , g ) ∈  H 1 ( R d ) ∩ L 1 ( R d )  ×  L 2 ( R d ) ∩ L 1 ( R d )  (8.1) further satisfy af ∈ L 1 ( R d ) ∩ L 2 ( R d ) , (8.2) then ther e exists a unique we ak solution u ∈ L ∞  (0 , ∞ ); H 1 ( R d )  ∩ W 1 , ∞  (0 , ∞ ); L 2 ( R d )  (8.3) to (1.1) (in the sense of (8.6) b elow) satisfying ∥ u ( t ) ∥ L 2 ≤ C I 00 ⟨ t ⟩ − 1 2 , ∥ ∂ t u ( t ) ∥ L 2 + ∥∇ u ( t ) ∥ L 2 ≤ C I 00 ⟨ t ⟩ − 1 , (8.4) wher e C > 0 and I 2 00 = ∥ f ∥ 2 L 2 + ∥∇ f ∥ 2 L 2 + ∥ af ∥ 2 L 1 + ∥ af ∥ 2 L 2 + ∥ g ∥ 2 L 1 + ∥ g ∥ 2 L 2 . (8.5) Recall that our main Theorem 1.2 concerns a solution of (1.1) constructed by means of the semigroup. Ikehata and T akeda, on the other hand, study weak solutions of (1.1) in [30] (with Ω = R d ), i.e. functions u : [0 , ∞ ) × Ω → C such that, for an y φ ∈ C ∞ c ([0 , ∞ ) × Ω), w e ha v e the follo wing iden tit y , where in particular all the integrals ha v e to b e ﬁnite: Z ∞ 0 Z Ω u ( t, x )  ∂ tt φ ( t, x ) − a ( x ) ∂ t φ ( t, x ) − (∆ − q ( x )) φ ( t, x )  d x d t = Z Ω g ( x ) φ (0 , x ) d x − Z Ω f ( x ) ∂ t φ (0 , x ) d x + Z Ω a ( x ) f ( x ) φ (0 , x ) d x. (8.6) This identit y arises by formally transp orting all (time and space) deriv ativ es to the test function φ . It turns out that, for the initial conditions we consider, every mild solution has the time-space regularity (8.3) and is a weak solution of (1.1). Lemma 8.2. L et the assumptions of The or em 1.2 hold. F or F ∈ K , let U ( t ) = ( u ( t ) , ∂ t u ( t )) = e t A F = e t A K F b e the unique mild solution of (2.1) in H and K . Then u ( t ) satisﬁes (8.3) (with R d r eplac e d by Ω ) and is a we ak solution of (1.1) ac c or ding to (8.6) . Pr o of. Fix F = ( f , g ) ∈ K . By Theorem 1.2, we ha v e ∥ u ( t ) ∥ H 1 + ∥ ∂ t u ( t ) ∥ L 2 ≲ ⟨ t ⟩ − 1 2 ∥ F ∥ K ≲ 1 , t ≥ 0 , and thus (8.3). It remains to prov e (8.6). Note ﬁrst that b y the prop erties of a mild solution, U : [0 , ∞ ) → K is contin uous. Hence u : [0 , ∞ ) → D t is contin uous by Prop osition 2.6. Setting w ( t ) := Z t 0 u ( s ) d s, t ≥ 0 , it follows that w ∈ C 1 ([0 , ∞ ); D t ) , ∂ t w ( t ) = u ( t ) , (8.7) and from the second comp onent of (2.8) w e get ∂ t u ( t ) = (∆ − q ) w ( t ) − a ( u ( t ) − f ) + g , t ≥ 0 . (8.8) Consider φ ∈ C ∞ c ([0 , ∞ ) × Ω; R ) (the identit y for φ ∈ C ∞ c ([0 , ∞ ) × Ω) follows b y considering Re φ and Im φ separately) and let T > 0 and Σ ⊂ R d b e op en and 38 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL b ounded with Σ ⊂ Ω suc h that supp φ ⊂ [0 , T ) × Σ. By (2.9) and (8.8), we ha v e Z T 0 Z Σ u ( t, x ) ∂ tt φ ( t, x ) d x d t = Z T 0 ⟨ u ( t ) , ∂ tt φ ( t ) ⟩ L 2 d t = − Z T 0 ⟨ ∂ t u ( t ) , ∂ t φ ( t ) ⟩ L 2 d t − ⟨ u (0) , ∂ t φ (0) ⟩ L 2 = − Z T 0 ⟨ (∆ − q ) w ( t ) − a ( u ( t ) − f ) + g , ∂ t φ ( t ) ⟩ L 2 d t − ⟨ f , ∂ t φ (0) ⟩ L 2 . (8.9) F rom a 1 2 ∈ L 2 loc (Ω) and the contin uit y of u : [0 , ∞ ) → D t , we conclude sup t ∈ [0 ,T ] ∥ au ( t ) ∥ L 1 (Σ) < ∞ , af = au (0) ∈ L 1 (Σ) . (8.10) By (8.7) and (8.10), for ﬁxed t ∈ [0 , T ] w e can write  (∆ − q ) w ( t ) − a ( u ( t ) − f ) , ∂ t φ ( t )  L 2 =  (∆ − q ) w ( t ) − a ( u ( t ) − f ) , ∂ t φ ( t )  D ∗ t ×D t =  (∆ − q ) w ( t ) , ∂ t φ ( t )  W ∗ ×W − Z Σ a ( x ) u ( t, x ) ∂ t φ ( t, x ) d x + Z Σ a ( x ) f ( x ) ∂ t φ ( t, x ) d x. (8.11) Using (8.10), w e can split the time integration in the last line of (8.9) according to the summands in (8.11); recall the iden tit y (8.8) and that ∂ t u ∈ C ([0 , ∞ ); L 2 (Ω)) b y the properties of our mild solution. F or the ﬁrst summand, by (2.20), the Green F ormula, in tegrating b y parts in time and using (8.7), we obtain Z T 0  (∆ − q ) w ( t ) , ∂ t φ ( t )  W ∗ ×W d t = − Z T 0  ⟨∇ w ( t ) , ∇ ∂ t φ ( t ) ⟩ L 2 + ⟨ w ( t ) , q ∂ t φ ( t ) ⟩ L 2  d t = Z T 0 ⟨ w ( t ) , ∂ t ((∆ − q ) φ ( t )) ⟩ L 2 d t = − Z T 0 ⟨ u ( t ) , (∆ − q ) φ ( t ) ⟩ L 2 d t. (8.12) Finally , since af + g ∈ L 1 (Σ) (see (8.10)), applying dominated conv ergence to exc hange the integral and time deriv ative leads to Z T 0 Z Σ  a ( x ) f ( x ) + g ( x )  ∂ t φ ( t, x ) d x d t = Z T 0 d d t Z Σ  a ( x ) f ( x ) + g ( x )  φ ( t, x ) d x d t = − Z Σ  a ( x ) f ( x ) + g ( x )  φ (0 , x ) d x. The proof is completed b y putting together the ab o v e equation with (8.9), a v ersion of (8.11) where the summands are integrated o v er t ∈ [0 , T ], and (8.12). □ Emplo ying the ab o v e lemma, it follows that our decay result con tains Theo- rem 8.1. T o prov e this, the Sob olev inequality is used. This indicates the origin of the diﬃculty in treating lo w dimensions d = 1 , 2 by the method in [30]. SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 39 Prop osition 8.3. L et the assumptions of The or em 8.1 hold and let F = ( f , g ) b e as in (8.1) and (8.2) . Then F ∈ K with ∥ F ∥ K ≲ I 00 and The or em 1.2 applies. The solution u ( t ) in The or em 1.2 c oincides with the unique we ak solution of (1.1) in (8.3) . In p articular, (1.9) , (1.10) and (1.11) imply (8.4) . Pr o of. Clearly Ω = R d with d ≥ 3, q ( x ) = 0 and 0 < a 0 ≤ a ( x ) ∈ C ( R d ) satisfy the basic assumptions of Theorem 1.2. F or F = ( f , g ) as in (8.1) and (8.2), we hav e F ∈ D t ⊕ L 2 ( R d ); notice that f ∈ Dom( a ) ⊂ Dom  a 1 2  . Moreov er, from d ≥ 3 and a straightforw ard application of H¨ older’s inequalit y it follows that h := af + g ∈ L 1 ( R d ) ∩ L 2 ( R d ) ⊂ L 2 d d +2 ( R d ) , ∥ h ∥ L 2 d d +2 ≤ ∥ h ∥ 2 d L 1 ∥ h ∥ d − 2 d L 2 . Com bining H¨ older’s inequalit y and the Sob olev inequality [1, Thm. 4.31] with the ab o v e, w e further estimate     Z R d h ( x ) ϕ ( x ) d x     ≤ ∥ h ∥ L 2 d d +2 ∥ ϕ ∥ L 2 d d − 2 ≲ ∥ h ∥ L 2 d d +2 ∥∇ ϕ ∥ L 2 ≲ ∥ h ∥ 2 d L 1 ∥ h ∥ d − 2 d L 2 ∥ ϕ ∥ W , for all ϕ ∈ C ∞ c ( R d ). This implies that h ∈ W ∗ and by Y oung’s inequalit y ∥ h ∥ W ∗ ≲ ∥ h ∥ 2 d L 1 ∥ h ∥ d − 2 d L 2 ≲ ∥ h ∥ L 1 + ∥ h ∥ L 2 . In particular, considering (1.7), we hav e F ∈ K . Finally , returning to h = af + g and emplo ying the triangle inequalit y , we arriv e at ∥ F ∥ K ≲ I 00 (see (1.8) and (8.5)). □ Appendix A. Description of the sp ace W The space W was deﬁned in the introduction as the Hilb ert space completion of C ∞ c (Ω) w.r.t. the p olar form of ∥ f ∥ 2 W = ∥∇ f ∥ 2 L 2 + ∥ q 1 2 f ∥ 2 L 2 . Th us, an elemen t of W is an equiv alence class of Cauc hy sequences in C ∞ c (Ω) for the norm ∥·∥ W . If { f n } n is a represen tativ e of this equiv alence class, then {∇ f n } n and { q 1 2 f n } n are Cauch y sequences in L 2 (Ω), and w e deﬁne ∇ f and q 1 2 f as their resp ectiv e limits. These deﬁnitions do not dep end on the choice of the representativ e { f n } n . Then we can deﬁne ( − ∆ + q ) ∈ B ( W , W ∗ ) by (( − ∆ + q ) f , g ) W ∗ ×W = ⟨∇ f , ∇ g ⟩ L 2 + ⟨ q 1 2 f , q 1 2 g ⟩ L 2 , f , g ∈ W . First recall that if (2.3) holds, then W = H 1 0 (Ω) ∩ Dom( q 1 2 ), while in general W need not b e a subspace of L 2 (Ω). Our goal is to identify every element of W with a function f ∈ L 2 loc (Ω), so that ∇ f and q 1 2 f can be understo o d in the sense of distributions. How ever, if q = 0 a.e. on a connected comp onen t of Ω, then f is only determined up to a constant thereon. W e show that W is included in a suitable quotien t space. T o this end, C q shall denote the space of lo cally constan t functions w (i.e. constant on ev ery connected comp onen t of Ω) such that q w = 0 a.e. on Ω. Prop osition A.1. W is a close d subsp ac e of ˙ D q := D q  C q , D q := n f ∈ L 1 loc (Ω) : ∇ f ∈ L 2 (Ω) d , q 1 2 f ∈ L 2 (Ω) o . The action of ∆ − q is then well-deﬁned on ˙ D q (and thus on W ) in the usual distributional sense. T o prov e the prop osition, we sho w that ˙ D q is a Hilb ert space and that the completion W can b e embedded in it. W e will use the following standard result. 40 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL Lemma A.2. L et ω  = ∅ b e an op en c onne cte d subset of R d . Ther e exists a non- de cr e asing (for inclusion) se quenc e { U j } j ∈ N of op en and c onne cte d Lipschitz subsets of ω such that S j ∈ N U j = ω . Pr o of. F or x ∈ R d and j ∈ N , w e set c j x = [ x 1 , x 1 + 2 − j ] × · · · × [ x d , x d + 2 − j ]. Fix x 0 ∈ ω and deﬁne C j =   [ x ∈ X j c j x   ◦ , X j =  x ∈ 2 − j Z d : c j x ⊂ ω ∩ B ( x 0 , j )  . F or j ≥ j 0 , where j 0 ∈ N is large enough (such that x 0 ∈ C j ), we denote b y U j the connected component of C j whic h con tains x 0 . Note that U j is open and Lipschitz and that U j ⊂ U j +1 for all j ≥ j 0 . Consider x ∈ ω arbitrary . Since ω is op en and connected, there is γ ∈ C ([0 , 1]; ω ) with γ (0) = x 0 and γ (1) = x . There exists j ≥ j 0 suc h that B ( γ ( t ) , 2 − j √ d ) ⊂ ω ∩ B ( x 0 , j ) , t ∈ [0 , 1] . W e set C j,γ =   [ y ∈ X j,γ c j y   ◦ , X j,γ =  y ∈ X j : c j y ∩ γ ([0 , 1])  = ∅  . Then γ ([0 , 1]) ⊂ C j,γ ⊂ C j . This implies that x is in the same connected component of C j as x 0 , so x ∈ U j . Finally , we ha ve S j ≥ j 0 U j = ω and the lemma is prov ed. □ The next lemma generalizes [26, Lem. I I.6.2]. Lemma A.3. The sp ac e ˙ D q is Hilb ert when e quipp e d with the p olar form of ∥ [ f ] ∼ ∥ ˙ D q := ∥ f ∥ W . Pr o of. F rom the prop erties of ∥ · ∥ W , it follo ws that ˙ D q is an inner pro duct space. Note therefore that if ∥ f ∥ W = 0 then in particular ∇ f = 0, so f is lo cally constant. W e also hav e q 1 2 f = 0, so q f = 0 and hence f ∈ C q . Positiv e deﬁniteness in ˙ D q is implied. F or the completeness, consider a Cauch y sequence { [ f n ] ∼ } n ⊂ ˙ D q , represen ted b y a sequence { f n } n ⊂ D q b eing Cauch y w.r.t. ∥ · ∥ W . T o pro v e the claim, one needs to ﬁnd a limit f ∈ D q , lim n →∞ ∥ f n − f ∥ W → 0 . By the completeness of L 2 -spaces, there exist g ∈ L 2 (Ω) d and h ∈ L 2 (Ω) suc h that lim n →∞ ∥∇ f n − g ∥ L 2 = 0 , lim n →∞ ∥ q 1 2 f n − h ∥ L 2 = 0 , (A.1) (implying the resp ectiv e conv ergence in D ′ (Ω) d and D ′ (Ω)). Let ω b e a connected comp onen t of Ω and { U j } j the sequence given by Lemma A.2. F or every j ∈ N , since  ∇ f n | U j  n is Cauc hy in L 2 ( U j ) and U j is b ounded, Lipsc hitz and connected, it follows from the P oincar ´ e–Wirtinger inequality and a completeness argument that there exists v j ∈ L 2 ( U j ) satisfying lim n →∞ ∥ ( f n − f n j ) − v j ∥ L 2 ( U j ) = 0 , f n j = 1 | U j | Z U j f n ( x ) d x. As this implies conv ergence in D ′ ( U j ), w e get ∇ v j = g | U j in the sense of distribu- tions. Using that q ∈ L 1 loc (Ω), we also ha v e lim n →∞ q 1 2 ( f n − f n j ) = q 1 2 v j in D ′ ( U j ) . (A.2) SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 41 Assume that q is not a.e. zero on ω ; w.l.o.g. we can assume that it is not a.e. zero on U 1 (hence on any U j , j ∈ N ). F rom (A.1) and (A.2), it follows that the sequence of scalars { f n j } n con verges to some c j ∈ C satisfying c j q 1 2 = h − q 1 2 v j on U j . Setting ϕ j = v j + c j w e hav e on U j ∇ ϕ j = g and q 1 2 ϕ j = h. Let j, k ∈ N with j ≤ k . Since ∇ ( ϕ j − ϕ k ) = 0 on U j , there exists c ∈ C such that ϕ j − ϕ k | U j = c . On the other hand, q 1 2 ( ϕ j − ϕ k ) = 0 on U j , so c = 0. Then ϕ k coincides with ϕ j on U j and we can deﬁne a function f ω b y f ω | U j = ϕ j , j ∈ N . No w assume that q = 0 on ω . In this case we set c 1 = 0. Then, as abov e, we see b y induction in j ∈ N that there exists c j +1 ∈ C such that v j +1 | U j + c j +1 = v j + c j . Again we set ϕ j = v j + c j and we can consider the unique function f ω whic h agrees with ϕ j on U j for all j ∈ N . In any case, we hav e a function f ω ∈ L 2 loc ( ω ) such that ∇ f n → ∇ f ω and q 1 2 f n → q 1 2 f ω in L 2 loc ( ω ). W e pro ceed similarly on all connected comp onen ts of Ω. This deﬁnes a function f ∈ L 2 loc (Ω) such that ∇ f n → ∇ f and q 1 2 f n → q 1 2 f in L 2 loc (Ω). By (A.1), w e deduce that ∇ f = g and q 1 2 f = h b elong to L 2 (Ω), so f ∈ D q and [ f n ] ∼ → [ f ] ∼ in ˙ D q as n → ∞ . □ Pr o of of Pr op osition A.1. The space C ∞ c (Ω) can b e understo od as a subspace of ˙ D q b y means of the linear injection ( C ∞ c (Ω) → ˙ D q f 7→ [ f ] ∼ . The completion W can be then iden tiﬁed with the closure of this subspace in ˙ D q . □ Appendix B. Sep ara tion of v ariables Lemma B.1. L et A and B b e two line ar op er ators in a Hilb ert sp ac e H such that A ⊂ B . Assume further that A − 1 and B − 1 exist and ar e everywher e deﬁne d on H . Then A = B . Pr o of. Let x ∈ Dom( B ) and set y := B x . Then x ′ := A − 1 y ∈ Dom( A ) and Ax ′ = y = B x . Since w e also ha v e Ax ′ = B x ′ (b ecause B extends A by assumption) and B is injectiv e, it follows that x ′ = x , i.e. x ∈ Dom( A ). □ Lemma B.2. L et H , H 1 , H 2 b e sep ar able Hilb ert sp ac es and U : H 1 ⊗ H 2 → H a unitary op er ator. L et B b e a line ar op er ator in H and assume further that: (i) ther e exist line ar op er ators B 1 , B 2 in H 1 , H 2 , r esp e ctively, such that B 1 ⊗ I H 2 + I H 1 ⊗ B 2 ⊂ U − 1 B U, (B.1) wher e the left hand side is deﬁne d on the line ar sp an of simple tensors in Dom( B 1 ) ⊗ Dom( B 2 ) ; (ii) ther e exists an orthonormal b asis { e j } j ∈ N ⊂ H 2 of eigenve ctors of B 2 with c orr esp onding eigenvalues { ζ j } j ∈ N ; (iii) ther e exists λ 0 ∈ ρ ( B ) such that λ 0 ∈ \ j ∈ N ρ ( B 1 + ζ j ) , sup j ∈ N ∥ ( B 1 + ζ j − λ 0 ) − 1 ∥ B ( H 1 ) < ∞ . (B.2) 42 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL Consider the line ar op er ator Dom( A ) :=  f = X j ∈ N f j ⊗ e j ∈ H 1 ⊗ H 2 : f j ∈ Dom( B 1 ) , j ∈ N , X j ∈ N ∥ ( B 1 + ζ j ) f j ∥ 2 H 1 < ∞  , Af := X j ∈ N (( B 1 + ζ j ) f j ) ⊗ e j , acting in H 1 ⊗ H 2 . Then A = U − 1 B U and ∥ ( B − λ 0 ) − 1 ∥ B ( H ) = sup j ∈ N ∥ ( B 1 + ζ j − λ 0 ) − 1 ∥ B ( H 1 ) . (B.3) Mor e over, in this c ase, for any λ ∈ C one has λ ∈ ρ ( B ) if and only if λ satisﬁes (B.2) (with λ 0 = λ ) and then formula (B.3) holds (with λ 0 = λ ). Pr o of. It is evident from the deﬁnition of A that the ﬁnite sums f N := N X j =1 f j ⊗ e j , f j ∈ Dom( B 1 ) , j = 1 , . . . , N , N ∈ N , form a core of A . The inclusion A ⊂ U − 1 B U then readily follo ws since U − 1 B U is closed, the f N lie in its domain (they are in the domain of the tensor in (B.1)) and Af N = N X j =1 (( B 1 + ζ j ) f j ) ⊗ e j = N X j =1 ( B 1 f j ) ⊗ e j + N X j =1 f j ⊗ ( ζ j e j ) = U − 1 B U f N . By assumption, { e j } j ∈ N is an orthonormal basis in H 2 and hence we can write an y g ∈ H 1 ⊗ H 2 as g = X j ∈ N g j ⊗ e j , g j ∈ H 1 , ∥ g ∥ 2 H 1 ⊗H 2 = X j ∈ N ∥ g j ∥ 2 H 1 , (see e.g. [47, Prop. I I.4.2] and note that the g j are uniquely determined). Let us deﬁne the op erator C λ 0 g := X j ∈ N (( B 1 + ζ j − λ 0 ) − 1 g j ) ⊗ e j , g ∈ H 1 ⊗ H 2 . It is clear from (B.2) that C λ 0 is w ell-deﬁned and b ounded on H 1 ⊗ H 2 . Moreov er, w e hav e C λ 0 g ∈ Dom( A ) for every g ∈ H 1 ⊗ H 2 and ( A − λ 0 ) C λ 0 g = X j ∈ N g j ⊗ e j = g . Similarly , for g ∈ Dom( A ) we ha v e C λ 0 ( A − λ 0 ) g = X j ∈ N g j ⊗ e j = g . It follows that λ 0 ∈ ρ ( A ) with ( A − λ 0 ) − 1 = C λ 0 . By Lemma B.1, we conclude A = U − 1 B U and hence ( B − λ 0 ) − 1 = U ( A − λ 0 ) − 1 U − 1 = U C λ 0 U − 1 . The equality (B.3) then follo ws since for any g ∈ H 1 ⊗ H 2 w e hav e ∥ C λ 0 g ∥ 2 H 1 ⊗H 2 = X j ∈ N ∥ ( B 1 + ζ j − λ 0 ) − 1 g j ∥ 2 H 1 . After having established the unitary equiv alence of A and B , the ﬁnal claim follo ws by repeating the abov e argument with λ 0 = λ . □ SEMIGROUP DECA Y FOR UNBOUNDED DAMPING 43 T o justify (7.6) for a ﬁxed λ ∈ C \ ( −∞ , 0], we apply Lemma B.2. W e b egin by noting that U : ( L 2 ( R ) ⊗ L 2 ( − 1 , 1) → L 2 (Ω) f ⊗ g 7→ f ( x ) g ( y ) determines a unique unitary op erator from H 1 ⊗ H 2 := L 2 ( R ) ⊗ L 2 ( − 1 , 1) onto H := L 2 (Ω) (see e.g. [47, Thm. I I.10 (a)]). T ake B := T λ as in (7.5) and B 1 := − ∂ 2 x + λx 2 n + λ 2 , Dom( B 1 ) := H 2 ( R ) ∩ Dom( x 2 n ) , B 2 := − ∂ 2 y , Dom( B 2 ) := H 2 ( − 1 , 1) ∩ H 1 0 ( − 1 , 1) . One can then verify the inclusion in Lemma B.2 (i). Indeed, for simple tensors f ⊗ g with f ∈ Dom( B 1 ) and g ∈ Dom( B 2 ), one can prov e that U ( f ⊗ g ) ∈ Dom( T λ ) = H 2 (Ω) ∩ H 1 0 (Ω) ∩ Dom  x 2 n  and moreov er that T λ U ( f ⊗ g ) =  − f ′′ + λx 2 n f + λ 2 f  ( x ) g ( y ) + f ( x )( − g ′′ )( y ) = U (( B 1 f ) ⊗ g + f ⊗ ( B 2 g )) . This extends to the linear span of simple tensors and assumption (i) follows. W e also recall that B 2 has a complete orthonormal set of eigenv ectors e j ( y ) = c j      cos  π 2 j y  , j = 1 , 3 , 5 , . . . , sin  π 2 j y  , j = 2 , 4 , 6 , . . . , with corresp onding eigen v alues ζ j =  j π 2  2 , j ∈ N . Finally , to v erify the assumption (iii), consider ﬁrst the case Im λ ≥ 0. By simple geometric considerations, one sees that Num( T λ ) ∪ [ j ∈ N Num( B 1 + ζ j ) ⊂  z ∈ C : Im z ≥ Im( λ 2 )  . It follows that all µ ∈ C with Im µ < Im( λ 2 ) are in the resolv ent set of T λ and B 1 + ζ j for all j ∈ N . Moreo v er, for such µ one has the resolven t b ound ∥ ( B 1 + ζ j − µ ) − 1 ∥ B ( L 2 ( R )) ≤ 1 Im( λ 2 ) − Im µ . Hence the assumption (iii) holds (uniformly) on any half plane Im µ ≤ Im( λ 2 ) − ε with ε > 0. An analogous claim is true in the case Im λ < 0. F rom the application of Lemma B.2 it follows that T λ is unitarily equiv alent to an inﬁnite direct sum of op erators U − 1 T λ U = M j ∈ N ( B 1 + ζ j ) ⊗ I span { e j } acting in the direct sum of spaces M j ∈ N L 2 ( R ) ⊗ span { e j } = L 2 ( R ) ⊗ L 2 ( − 1 , 1) . F rom this, how ev er, (7.6) follows immediately since L 2 ( R ) ⊗ span { e j } ≃ L 2 ( R ) , ( B 1 + ζ j ) ⊗ I span { e j } ≃ B 1 + ζ j = T λ,j . In the ab o v e, for linear operators T 1 and T 2 acting in Hilb ert spaces H 1 and H 2 , resp ectiv ely , we write H 1 ≃ H 2 if there exists a unitary op erator V : H 1 → H 2 , as 44 ANTONIO ARNAL, BORBALA GERHA T, JULIEN ROYER, AND PETR SIEGL w ell as T 1 ≃ T 2 if moreov er V T 1 V − 1 = T 2 . Note that the resolven t form ula (B.3) no w translates into (7.13). References [1] Adams, R. A., and Fournier, J. J. F. Sob olev sp ac es , 2nd ed. Elsevier, Amsterdam, 2003. [2] Aloui, L., Ibrahim, S., and Khenissi, M. 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(PS, AA) Institute of Applied Ma thema tics, Graz University of Technology, Steyr- ergasse 30, 8010 Graz, Austria Email address : siegl@tugraz.at Email address : aarnalperez01@qub.ac.uk (BG) Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneu- burg, Austria Email address : borbala.gerhat@ista.ac.at (JR) Institut de Ma th ´ ema tiques de Toulouse, Universit ´ e de Toulouse, 118 route de Narbonne, 31062 Toulouse C ´ edex 9, France Email address : julien.royer@math.univ-toulouse.fr

Semigroup decay for the wave equation with unbounded damping

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