A Class of Degenerate Hyperbolic Equations with Neumann Boundary Conditions and Its Application to Observability

A Class of Degenerate Hyp erb olic Equations with Neumann Boundary Conditions and Its Application to Observ abilit y Dong-Hui Y ang and Jie Zhong Marc h 31, 2026 Abstract W e establish a mixed observ ability inequalit y for a class of degenerate hyperb olic equations on the cylindrical domain Ω = T × (0 , 1) with mixed Neumann–Dirichlet b oundary conditions. The degeneracy acts only in the radial v ariable, whereas the p e- rio dic angular v ariable allo ws propagation with a strong tangen tial component, making a direct top-b oundary observ ation delicate. F or α ∈ [1 , 2), we prov e that the solution can b e con trolled b y a b oundary observ ation on the top b oundary together with an in terior observ ation on a narrow strip. The pro of com bines a w eigh ted functional frame- w ork, improv ed regularity , a cut-off decomp osition in the angular v ariable, a m ultiplier argumen t for the lo calized comp onent, and an energy estimate for the remainder. Keyw ords. Degenerate hyperb olic equations; observ abilit y inequalit y; Neumann b oundary conditions; m ultiplier metho d; weigh ted Sob olev spaces. 2020 Mathematics Sub ject Classification. 35L80, 93B07, 35L05, 35L20. Con ten ts 1 In tro duction 3 2 Preliminary results 6 2.1 Solution spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Green’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Neumann b oundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 The sp ectrum for the partial differential op erator A . . . . . . . . . . . . . . 18 2.5 Sp ectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Existence and uniqueness of weak solutions to equation (1.1) . . . . . . . . . 25 3 Observ abilit y 35 3.1 A motiv ating example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Cut-off decomp osition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Preliminary identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Multiplier argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1 In tro duction This pap er is concerned with the observ abilit y of a degenerate hyperb olic equation p osed on the cylindrical domain Ω = T × (0 , 1), in which the degeneracy acts only in the radial direction while the angular v ariable remains p erio dic. In the uniformly h yp erb olic setting, observ abilit y and con trollability are link ed by the Hilb ert Uniqueness Metho d (HUM) of J.- L. Lions [ 17 ]; see also [ 5 , 6 , 9 , 11 , 15 , 16 , 18 , 19 , 23 , 25 ]. F or degenerate h yp erb olic equations, ho wev er, the geometry of propagation and the functional framework b ecome considerably subtler. Degenerate hyperb olic equations hav e b een studied in sev eral works [ 1 , 3 , 10 , 13 , 21 , 24 ], but the av ailable theory is still largely one-dimensional, with [ 21 ] as a notable higher- dimensional exception. In that w ork, a cut-off metho d was used to establish controllabilit y when the control acts near the degenerate region. The same lo calization philosoph y has also app eared in related problems [ 21 , 22 ]. In particular, [ 22 ] used the cut-off metho d in the construction of the Diric hlet map for non-homogeneous degenerate h yp erb olic equations. These preceden ts suggest that, in higher-dimensional degenerate geometries, lo calization is not merely a technical conv enience but often part of the correct analytical structure. It is natural to ask whether one can observ e the system from the nondegenerate part of the b oundary . In the cylindrical geometry studied here, this question b ecomes substan tially more delicate than in the one-dimensional setting. In fact, the p erio dic v ariable θ allows propagation with a strong angular comp onent, while the degeneracy acts only in the radial direction. This mismatc h b etw een geometry and observ ation is the source of the main difficult y: a direct top-b oundary observ ation do es not isolate the angular comp onent in any ob vious wa y . The main contribution of this pap er is to show that, in the range α ∈ [1 , 2), the ob- serv abilit y problem can still b e resolved after a suitable lo calization in the angular v ariable. The key p oint is not to eliminate the angular propagation, but to separate it analytically . W e decomp ose the solution in to a comp onent supp orted aw ay from a narro w angular strip and a complementary comp onent concentrated near that strip. The lo calized comp onent is handled by a multiplier argument on a single co ordinate c hart, whereas the remainder is con trolled by an energy estimate. As a consequence, the final observ abilit y inequality naturally tak es a mixed form: a b oundary observ ation on the top b oundary Γ = T × { 1 } together with an interior observ ation on the strip ω . F rom this p ersp ective, the strip ω is not an auxiliary artifact of the pro of. It is forced b y the geometry of the problem and b y the quasimo de-type obstruction exhibited in Section 3 . That mechanism explains wh y a direct top-b oundary observ ation is not the right ob ject to isolate the angular comp onent, and why a lo calized decomp osition is the natural replacement. The presen t mo del therefore pro vides a concrete higher-dimensional setting in whic h ob- serv abilit y can b e established for a degenerate hyperb olic equation with Neumann b oundary conditions. T o the b est of our kno wledge, suc h results do not seem to b e a v ailable in the existing literature. 3 A further difficult y comes from regularity . In degenerate h yp erb olic problems, weak solu- tions do not automatically p ossess the level of smo othness needed to justify the in tegrations b y parts required by m ultiplier arguments, esp ecially near the degenerate b oundary . Thus an y observ abilit y pro of in this setting must also address the underlying regularity mec hanism. F or degenerate parabolic equations, improv ed regularit y estimates ha ve b een used suc- cessfully in controllabilit y problems; see, for instance, [ 2 ]. The hyperb olic setting is subtler, b oth b ecause the propagation mechanism is differen t and b ecause the c hoice of a useful m ultiplier is less rigid. In the present cylindrical geometry , ho wev er, the regularity of weak solutions can still b e impro ved enough to justify the m ultiplier argumen t. This extra reg- ularit y is an essential part of the pro of, not a purely auxiliary ingredien t. W e also exp ect that the same strategy should remain meaningful when T in Assumption 1.1 is replaced b y a smo oth manifold without b oundary . W e consider the observ abilit y of the follo wing degenerate hyperb olic equation:            ∂ tt ϕ − div( A ∇ ϕ ) = f , in Q, ϕ = 0 , on Γ × (0 , T ) , ∂ φ ∂ ν A = 0 , on Γ ∗ × (0 , T ) , ϕ (0) = ϕ 0 , ∂ t ϕ (0) = ϕ 1 , in Ω , (1.1) where A, Ω , Q and Γ , Γ ∗ , and ϕ 0 , ϕ 1 and f are defined in Assumption 1.1 . Assumption 1.1 (Assumptions and Notations) . Let N = { 0 , 1 , 2 , · · · } and N ∗ = { 1 , 2 , · · · } . Let Ω = T × (0 , 1) , Γ ∗ = T × { 0 } and Γ = T × { 1 } , and T > 0, and Q = Ω × (0 , T ), where T = { x ∈ R 2 : | x | = 1 } . Denote z = ( θ , r ) ∈ Ω. Let α ∈ [1 , 2) and w = r α , and A = diag (1 , w ) = diag (1 , r α ). Denote ν A = Aν on ∂ Q . Let ϕ 0 ∈ H 1 Γ (Ω; w ) , ϕ 1 ∈ L 2 (Ω) and f ∈ L 2 ( Q ). Here, the weigh t Sob olev space H 1 Γ (Ω; w ) will b e defined in ( 2.1 ). Remark 1.2. (1) Equation ( 1.1 ) is indeed a degenerate hyperb olic equation with perio dic b oundary conditions                  ∂ tt y ( θ , r, t ) − div ( A ∇ y ( θ , r, t )) = f , in (0 , 2 π ) × (0 , 1) × (0 , T ) , y (0 , r, t ) = y (2 π , r, t ) , for all ( r, t ) ∈ (0 , 1) × (0 , T ) , [ r α ∂ r y ( θ , r, t )] r =0 = 0 , for all ( θ , t ) ∈ (0 , 2 π ) × (0 , T ) , y ( θ , 1 , t ) = 0 , for all ( θ , t ) ∈ (0 , 2 π ) × (0 , T ) , y (0) = y 0 , ∂ t y = y 1 , in Ω . (2) W e explain the op erator ∂ θθ in the following: Let L 2 1 = span  f 1 n ≡ sin nθ  n ∈ N ∗ , L 2 2 = span  f 2 n ≡ cos nθ  n ∈ N , L 2 ( T ) = L 2 1 ⊕ L 2 2 . 4 Define H k ( T ) = ( f = X n ∈ N ∗ a 1 n f 1 n + X n ∈ N a 2 n f 2 n : ( a 2 0 ) 2 + X n ∈ N ∗ n 2 k  ( a 1 n ) 2 + ( a 2 n ) 2  < + ∞ ) with norm ∥ f ∥ 2 H k ( T ) = ( a 2 0 ) 2 + X n ∈ N n 2 k  ( a 1 n ) 2 + ( a 2 n ) 2  . F or eac h f = P n ∈ N ∗ a 1 n f 1 n + P n ∈ N a 2 n f 2 n , we hav e ∂ θθ : H k ( T ) → H k − 2 ( T ) , with ∂ θθ f = − X n ∈ N ∗ n 2 a 1 n f 1 n − X n ∈ N n 2 a 2 n f 2 n . Hence, we may regard ∂ θ as an op erator acting on θ ∈ [0 , 2 π ). (3) It is clear that the degenerate h yp erb olic equation ( 1.1 ) is degenerate on the b oundary Γ ∗ . Theorem 1.3. L et ϕ 0 ∈ H 1 Γ (Ω; w ) and ϕ 1 ∈ L 2 (Ω) . L et ϕ b e the solution of ( 1.1 ) with r esp e ct to ( ϕ 0 , ϕ 1 , f = 0) . If T > √ 2 / (2 − α ) , then ther e exists a c onstant C = C ( T , α, δ 0 ) > 0 such that h (2 − α ) T − √ 2 i E (0) ≤ C Z Z Γ × (0 ,T ) ( ∂ r ϕ ) 2 d S d t + C Z Z ω × (0 ,T )  ϕ 2 + ( ∂ t ϕ ) 2 + A ∇ ϕ · ∇ ϕ  d z d t, Remark 1.4. The app earance of the in terior term on ω is not an artifact of the pro of. As explained in Section 3 , the p erio dic v ariable θ allows profiles with strong angular oscillation and w eak radial p enetration, so that a purely top-b oundary observ ation is not the natural quan tity to con trol directly in the present argument. The cut-off decomp osition isolates this angular obstruction on the strip ω , and the final estimate should therefore be read as a mixed observ abilit y inequality: b oundary observ ation on Γ together with interior observ ation on ω . The pro of of Theorem 1.3 rests on three ingredien ts: a w eighted functional framework adapted to the mixed Neumann–Dirichlet geometry of ( 1.1 ), a cut-off decomp osition ϕ = ψ + ξ isolating the angular obstruction, and the combination of a m ultiplier estimate for ψ with an energy estimate for ξ . The pap er is organized as follows. In Section 2 we in tro duce the w eighted spaces, Green’s form ula, the Neumann trace, the sp ectral prop erties of the op erator A , and the w ell-p osedness theory for ( 1.1 ). In Section 3 we explain the quasimo de-t yp e obstruction that motiv ates the cut-off decomp osition, deriv e the multiplier identit y for ψ , estimate the lo calized remainder ξ , and complete the pro of of Theorem 1.3 . 5 2 Preliminary results In this section w e collect the analytic ingredients used later in the observ abilit y argumen t. W e introduce the weigh ted Sob olev spaces asso ciated with ( 1.1 ), establish Green’s formula and the Neumann trace in the presen t degenerate setting, describ e the spectral prop erties of the op erator A , and recall the well-posedness theory for ( 1.1 ). 2.1 Solution spaces W e b egin with the w eigh ted energy space H 1 (Ω; w ) =  u ∈ L 2 (Ω) : Z Ω ∇ u · A ∇ u d z < + ∞  . Its inner pro duct and norm are defined by ( u, v ) H 1 (Ω; w ) = Z Ω ( ∇ u · A ∇ v + uv )d z , ∥ u ∥ H 1 (Ω; w ) = ( u, u ) 1 2 H 1 (Ω; w ) for all u, v ∈ H 1 (Ω; w ). W e also in tro duce C ∞ Γ (Ω) = { u ∈ C ∞ (Ω) : there exists δ > 0 such that dist(supp u, Γ) > δ } , H 1 Γ (Ω) = the closure of the set C ∞ Γ (Ω) in H 1 (Ω) , H 1 Γ (Ω; w ) = the closure of the set C ∞ Γ (Ω) in H 1 (Ω; w ) , (2.1) where supp u denotes the supp ort of u . W e then define H − 1 Γ (Ω; w ) is the dual space of H 1 Γ (Ω; w ) with pivot L 2 (Ω) . Remark 2.1. It is clear that if u ∈ H 1 Γ (Ω) (or u ∈ H 1 Γ (Ω; w )), then u = 0 on Γ in the sense of the classical Sob olev trace. Next we set H 2 (Ω; w ) =  u ∈ H 1 (Ω; w ) : A u ∈ L 2 (Ω)  . Its inner pro duct and norm are defined by ( u, v ) H 2 (Ω; w ) = Z Ω ( A u )( A v )d z + ( u, v ) H 1 (Ω; w ) , ∥ u ∥ H 2 (Ω; w ) = ( u, u ) 1 2 H 2 (Ω; w ) for all u, v ∈ H 2 (Ω; w ). Finally , w e let D ( A ) = H 1 Γ (Ω; w ) ∩ H 2 (Ω; w ) . 6 It is well known that H 1 Γ (Ω; w ) , and H 1 (Ω; w ) , and H 2 (Ω; w ) are Hilb ert spaces; see [ 12 , 14 ]. Remark 2.2. The space C ∞ (Ω) is dense in H 2 (Ω; w ). In fact, H 2 (Ω) is contained in H 2 (Ω; w ) b ecause R Ω ∇ u · A ∇ u d z ≤ R Ω |∇ u | 2 d z , and Z Ω [ A u ] 2 d z = Z Ω  ∂ θθ u + αr α − 1 ∂ r u + r α ∂ rr u  2 d z ≤ C ∥ u ∥ 2 H 2 (Ω) b y α ∈ [1 , 2). On the other hand, since H 2 (Ω; w ) ⊆ H 2 loc (Ω), it follo ws that H 2 (Ω) is dense in H 2 (Ω; w ). Hence C ∞ (Ω) is dense in H 2 (Ω; w ). The following lemma is a v arian t of Hardy’s inequality . Lemma 2.3. (1) L et α ∈ (1 , 2) . Then ther e exists a c onstant C > 0 , dep ending only on α , such that Z Ω r α − 2 u 2 d z ≤ C Z Ω r α ( ∂ r u ) 2 d z for al l u ∈ H 1 Γ (Ω; w ) . (2) L et α = 1 . F or e ach β > 0 , λ > 0 , ther e exists a c onstant C > 0 , dep ending only on β and λ , such that λ Z Ω r − 1+ β u 2 d z ≤ C Z Ω r ( ∂ r u ) 2 d z for al l u ∈ H 1 Γ (Ω; w ) . Pr o of. By densit y , it suffices to pro ve the statement for u ∈ C ∞ Γ (Ω). (1) W e pro ve the case α ∈ (1 , 2). F rom u = 0 on Γ, w e get | u ( θ , r ) | 2 =  Z 1 r ∂ r u ( θ , s )d s  2 ≤  Z 1 r s α +1 2 | ∂ r u ( θ , s ) | 2 d s   Z 1 r s − α +1 2 d s  ≤ 2 α − 1 r − α +1 2 Z 1 r s α +1 2 | ∂ r u ( θ , s ) | 2 d s, and therefore Z 1 0 r α − 2 | u ( θ , r ) | 2 d r ≤ 2 α − 1 Z 1 0 Z 1 r r α − 3 2 s α +1 2 | ∂ r ( θ , s ) | 2 d s d r = 2 α − 1 Z 1 0 Z s 0 r α − 3 2 s α +1 2 | ∂ r u ( θ , s ) | 2 d r d s = 4 ( α − 1) 2 Z 1 0 s α | ∂ r u ( θ , s ) | 2 d s. 7 Hence Z Ω r α − 2 u 2 d z = Z T Z 1 0 r α − 2 u 2 d r d θ ≤ 4 ( α − 1) 2 Z T Z 1 0 r α | ∂ r u ( θ , r ) | 2 d r d θ . (2) W e pro ve the case α = 1. F rom R Ω r u 2 d z ≤ R Ω u 2 d z , and Z Ω   ∇ ( r u 2 )   d z = Z Ω   u 2 + 2 r u ∇ u   d z ≤ 5 Z Ω u 2 d z + 2 Z Ω ∇ u · A ∇ u d z , w e obtain r u 2 ∈ W 1 , 1 0 (Ω). Then Z Ω u 2 d z = Z Ω ∂ r  r u 2  d z − 2 Z Ω r u∂ r u d z = − 2 Z Ω r u∂ r u d z ≤ 1 2 Z Ω u 2 d z + 4 Z Ω r | ∂ r u | 2 d z , This implies that Z Ω u 2 d z ≤ 8 Z Ω r | ∂ r u | 2 d z . (2.2) No w, let β > 0 , λ > 0. F or each u ∈ H 1 Γ (Ω; w ), since for each δ ∈ (0 , 1 8 ) we hav e 0 ≤ Z { δ 0, we ha v e ( λ + 1) r β 2 − β 2 ≤ 0 for all 0 ≤ r ≤  β 2( λ + 1)  2 β . Denote A = min ( 1 ,  β 2( λ + 1)  2 β ) , C ( β , λ ) = sup r ∈ [0 , 1]  ( λ + 1) r β 2 − β 2  r − 1+ β 2  . Since A → 0 + as λ → + ∞ or as β → 0 + , and C ( β , λ ) → + ∞ as λ → + ∞ or as β → 0 + (tak e r = ( β λ +1 ) 2 β ), we obtain λ Z { δ 0 dep ending only on α . This is the Poincar ´ e inequalit y . Hence, in what follows, we use the inner pro duct and norm ( u, v ) H 1 Γ (Ω; w ) = Z Ω ∇ u · A ∇ v d z , ∥ u ∥ H 1 Γ (Ω; w ) = ( u, u ) 1 2 H 1 Γ (Ω; w ) on H 1 Γ (Ω; w ), resp ectiv ely . Lemma 2.5. The emb e dding H 1 Γ (Ω; w )  → L 2 (Ω) is c omp act. Pr o of. Let ∥ u n ∥ H 1 Γ (Ω; w ) ≤ M ( n ∈ N ∗ ) for some M > 0. Then there exists a subsequence of { u n } n ∈ N ∗ , still denoted by itself, and u ∗ ∈ H 1 Γ (Ω; w ) suc h that u n → u ∗ w eakly in H 1 Γ (Ω; w ). 9 W e will sho w that there exists a subsequence of { u n } n ∈ N ∗ , still denoted by itself, such that u n → u ∗ strongly in L 2 (Ω). Without loss of generality , we assume u ∗ = 0 (for otherwise, replacing u n b y u n − u ∗ for all n ∈ N ∗ ). (1) Let α ∈ (1 , 2). Let  > 0. Since for all n ∈ N ∗ w e hav e Z Ω δ u 2 n d z ≤ δ 2 − α Z Ω δ r α − 2 u 2 n d z ≤ δ 2 − α Z Ω ∇ u n · A ∇ u n d z ≤ M δ 2 − α b y Remark ( 2.4 ), choosing δ < δ 0 = ( 1 4 M  2 ) 1 2 − α , then R Ω δ 0 u 2 n d z < 1 4  2 for all n ∈ N ∗ . Note that u n | Ω δ 0 ∈ H 1 (Ω δ 0 ) for all n ∈ N ∗ , and H 1 (Ω δ 0 )  → L 2 (Ω δ 0 ) is compact, then there exists a subsequence, still denoted by itself, suc h that u n | Ω δ 0 → 0 strongly in L 2 (Ω δ 0 ). Hence there exists n 0 ∈ N ∗ suc h that for all n ≥ n 0 w e hav e ∥ u n | Ω δ 0 ∥ L 2 (Ω δ 0 ) < 1 2  , and this implies that R Ω u 2 d z <  . (2) Let α = 1. T aking β = 1 2 and λ = 1 in ( 2.3 ), then Z Ω r − 1 2 u 2 d z ≤ C Z Ω r ( ∂ r u ) 2 d z , where the constant C is absolute. By the same argumen t as (1), w e complete the proof of this lemma. 2.2 Green’s form ula Green’s formula is the foundation for p erforming in tegration by parts for solutions of ( 1.1 ). In this section, we establish this form ula. W e introduce the follo wing spaces H (div , Ω) =  u ∈ [ L 2 (Ω)] 2 : Z Ω u · u d z < + ∞ , Z Ω [div u ] 2 d z < + ∞  with inner pro duct ( u , v ) H (div , Ω) = Z Ω  u · v + [div u ][div v ]  d z ; Additional, we define H (div , Ω; w − 1 ) =  u ∈ [ L 2 (Ω)] 2 : Z Ω u · A − 1 u d z < + ∞ , Z Ω [div u ] 2 d z < + ∞  with inner pro duct ( u , v ) H (div , Ω; w − 1 ) = Z Ω  u · A − 1 v + [div u ][div v ]  d z . Lemma 2.6. The sp ac es ( H (div , Ω) , ( · , · ) H (div , Ω) ) and ( H (div , Ω; w − 1 ) , ( · , · ) H (div , Ω; w − 1 ) ) ar e Hilb ert sp ac es. 10 Pr o of. It is obvious that ( · , · ) H (div , Ω) and ( · , · ) H (div , Ω; w − 1 ) are bilinear functionals. Hence H (div , Ω) and H (div , Ω; w − 1 ) are inner pro duct spaces. Let { u n } n ∈ N b e a Cauc hy sequence in H (div , Ω). Then there exists u ∗ and v suc h that u n → u ∗ strongly in [ L 2 (Ω)] 2 , and div u → v strongly in L 2 (Ω) . Then for each ψ ∈ C ∞ 0 (Ω) we hav e Z Ω v ψ d z = lim n →∞ Z Ω [div u n ] ψ d z = − lim n →∞ Z Ω u n · ∇ ψ d z = − Z Ω u ∗ · ∇ ψ d z , i.e., v = div u ∗ in the sense of distribution. Hence v = div u ∗ in L 2 (Ω). Therefore, H (div , Ω) is a Hilb ert space. Let { u n } n ∈ N b e a Cauch y sequence in H (div , Ω; w − 1 ). Note that H (div , Ω; w − 1 )  → H (div , Ω) is contin uous, then there exists u ∗ suc h that u n → u ∗ strongly in H (div , Ω), i.e., u n → u ∗ strongly in [ L 2 (Ω)] 2 , and div u n → div u ∗ strongly in L 2 (Ω) . No w, since A − 1 2 u n → h in [ L 2 (Ω)] 2 for some h ∈ [ L 2 (Ω)] 2 , then for eac h Ξ ∈ [ C ∞ 0 (Ω)] 2 w e ha ve Z Ω h · Ξd z = lim n →∞ Z Ω A − 1 2 u n · Ξd z = lim n →∞ Z Ω u n · A − 1 2 Ξd z = Z Ω u ∗ · A − 1 2 Ξd z = Z Ω A − 1 2 u ∗ · Ξd z b y A − 1 2 Ξ ∈ [ C ∞ 0 (Ω)] 2 . Hence h = A − 1 2 u ∗ . This pro ves H (div , Ω; w − 1 ) is a Hilb ert space. Lemma 2.7. The sp ac e [ C ∞ (Ω)] 2 is dense in H (div , Ω) . However, [ C ∞ (Ω)] 2 is not dense in H (div , Ω; w − 1 ) . Pr o of. It is ob vious that the function u = (1 , 1) ∈ [ C ∞ (Ω)] 2 , but ∥ u ∥ H (div , Ω; w − 1 ) = Z Ω  1 + r − α  d z = + ∞ since α ∈ [1 , 2). W e no w pro ve that the space [ C ∞ (Ω)] 2 is dense in H (div , Ω). This result is classical and can b e found, for instance, in [ 7 , Theorem 1, Section 2, Chapter IX, Part A] or [ 20 , Chapter 20]. Definition 2.8. W e define γ ν : H (div , Ω) → H − 1 2 (Γ ∗ ) b y γ ν u = u · ν = − u 2 for all u = ( u 1 , u 2 ) ∈ [ C ∞ (Ω)] 2 , 11 where H − 1 2 (Γ ∗ ) is the dual space of H 1 2 (Γ ∗ ) with pivot space L 2 (Γ ∗ ). Remark 2.9. W e note that the normal trace op erator γ ν : H (div , Ω) → H − 1 2 (Γ) defined by γ ν u = u · ν = u 2 for all u = ( u 1 , u 2 ) ∈ [ C ∞ (Ω)] 2 is also well defined. Clearly , this case is the same as that in Definition 2.8 . Lemma 2.10. The op er ator γ ν is a b ounde d line ar op er ator. Pr o of. F or each u ∈ [ C ∞ (Ω)] 2 , we hav e ⟨ γ ν u , v ⟩ H − 1 2 (Γ ∗ ) ,H 1 2 (Γ ∗ ) = Z Γ ∗ ( u · ν ) v d S = Z Ω u · ∇ e v d z + Z Ω e v div u d z ≤ 3 ∥ u ∥ H 1 (div , Ω) ∥ e v ∥ H 1 Γ (Ω) for all v ∈ H 1 2 (Γ ∗ ), where e v ∈ H 1 Γ (Ω) is an extension of v (i.e., e v = v on Γ ∗ , the definition of H 1 Γ (Ω) is defined in ( 2.1 )), then    ⟨ γ ν u , v ⟩ H − 1 2 (Γ ∗ ) ,H 1 2 (Γ ∗ )    ≤ 3 ∥ u ∥ H 1 (div , Ω) inf {∥ e v ∥ H 1 Γ (Ω) : e v is an extension of v } ≡ 3 ∥ u ∥ H 1 (div , Ω) ∥ v ∥ H 1 2 (Γ ∗ ) . Hence ∥ γ ν u ∥ H − 1 2 (Γ ∗ ) ≤ 3 ∥ u ∥ H (div , Ω) . No w, from Lemma 2.7 , define γ ν u = lim n →∞ γ ν u n for [ C ∞ (Ω)] 2 ∋ u n → u strongly in H (div , Ω), then ∥ γ ν u ∥ H − 1 2 (Γ ∗ ) ≤ 3 ∥ u ∥ H 1 (div , Ω) , for all u ∈ H (div , Ω) . Since γ ν is obviously a linear op erator, we complete the pro of of this lemma. Lemma 2.11. F or al l u ∈ H (div , Ω) and al l v ∈ H 1 Γ (Ω) , we have Z Ω ([div u ] v + u · ∇ v ) d z = ⟨ γ ν u , v ⟩ H − 1 2 (Γ ∗ ) ,H 1 2 (Γ ∗ ) . 12 Pr o of. Let { u n } n ∈ N ∗ ⊆ [ C ∞ (Ω)] 2 suc h that u n → u strongly in H (div , Ω). Then ⟨ γ ν u , v ⟩ H − 1 2 (Γ ∗ ) ,H 1 2 (Γ ∗ ) = lim n →∞ ⟨ γ ν u n , v ⟩ H − 1 2 (Γ ∗ ) ,H 1 2 (Γ ∗ ) = lim n →∞ Z Γ ∗ ( γ ν u n ) v d S = lim n →∞ Z Ω ([div u n ] v + u n · ∇ v ) d z = Z Ω ([div u ] v + u · ∇ v ) d z for each v ∈ H 1 Γ (Ω). This completes the pro of of the lemma. No w, for each δ ∈ [0 , 1 8 ], as in Definition 2.8 , we can define γ ν : H (div , Ω) → H − 1 2 (Γ ∗ δ ) . Then, from Lemma 2.11 , we hav e ⟨ γ ν u , v ⟩ H − 1 2 (Γ ∗ ) ,H 1 2 (Γ ∗ ) − ⟨ γ ν u , v ⟩ H − 1 2 (Γ ∗ δ ) ,H 1 2 (Γ ∗ δ ) = Z Ω − Ω δ ([div u ] v + u · ∇ v ) d z for all u ∈ H (div , Ω) and v ∈ H 1 Γ (Ω). Hence lim δ → 0 + ⟨ γ ν u , v ⟩ H − 1 2 (Γ ∗ δ ) ,H 1 2 (Γ ∗ δ ) = ⟨ γ ν u , v ⟩ H − 1 2 (Γ ∗ ) ,H 1 2 (Γ ∗ ) for all u ∈ H (div , Ω) and v ∈ H 1 Γ (Ω). Lemma 2.12. F or every u ∈ H (div , Ω) and every v ∈ H 1 Γ (Ω) , the function f ( δ ; v ) = Z Γ ∗ δ v 2 d S, g ( δ ; u , v ) = ⟨ γ ν u , v ⟩ H − 1 2 (Γ ∗ δ ) ,H 1 2 (Γ ∗ δ ) with r esp e ct to δ ∈ [0 , 1 8 ] ar e c ontinuous. Pr o of. Let v ∈ H 1 Γ (Ω). Since for all 0 ≤ δ 1 ≤ δ 2 ≤ 1 8 w e hav e | f ( δ 1 ; v ) − f ( δ 2 ; v ) | =      Z Γ ∗ δ 1 v 2 d S − Z Γ ∗ δ 2 v 2 d S      =      Z Ω δ 1 − Ω δ 2 ∂ r v 2 d z      ≤ Z Ω δ 1 − Ω δ 2  v 2 + |∇ v | 2  d z = ∥ v ∥ 2 H 1 Γ (Ω δ 1 − Ω δ 2 ) , and then f ( δ ; v ) is a contin uous function with resp ect to δ ∈ [0 , 1 8 ]. 13 Let u ∈ H (div , Ω) and v ∈ H 1 Γ (Ω). Since for all 0 ≤ δ 1 ≤ δ 2 ≤ 1 8 w e hav e | g ( δ 1 ; u , v ) − g ( δ 2 ; u , v ) | =     ⟨ γ ν u , ν ⟩ H − 1 2 (Γ ∗ δ 1 ) ,H 1 2 (Γ ∗ δ 1 ) − ⟨ γ ν u , ν ⟩ H − 1 2 (Γ ∗ δ 2 ) ,H 1 2 (Γ ∗ δ 2 )     = Z Ω δ 1 − Ω δ 2 ([div u ] v + u · ∇ v ) d z ≤ 3 ∥ u ∥ H 1 (div , Ω δ 1 − Ω δ 2 ) ∥ v ∥ H 1 (Ω δ 1 − Ω δ 2 ) , and then g ( δ ; u , v ) is a contin uous function with resp ect to δ ∈ [0 , 1 8 ]. Next we establish the Green formula for u ∈ H (div , Ω; w − 1 ) and v ∈ H 1 Γ (Ω; w ), namely , Z Ω ([div u ] v + u · ∇ v ) d z = 0 . F or this, w e define T w : H (div , Ω; w − 1 ) × H 1 Γ (Ω; w ) → R , T w ( u , v ) = Z Ω ([div u ] v + u · ∇ v ) d z . (2.4) Lemma 2.13. The op er ator T w is a b ounde d biline ar functional. Pr o of. Clearly , T w is a bilinear functional. No w, for each u ∈ H (div , Ω; w − 1 ) and every v ∈ H 1 Γ (Ω; w ), w e hav e |T w ( u , v ) | =     Z Ω ([div u ] v + u · ∇ v ) d z     ≤  Z Ω [div u ] 2 d z  1 2  Z Ω v 2 d z  1 2 +  Z Ω u · A − 1 u d z  1 2  Z Ω ∇ v · A ∇ v d z  1 2 ≤ 3 ∥ u ∥ H (div , Ω; w − 1 ) ∥ v ∥ H 1 Γ (Ω; w ) , hence, T w is a b ounded bilinear functional. This completes the pro of of this lemma. Theorem 2.14. F or e ach u ∈ H (div , Ω; w − 1 ) , we have T w ( u , v ) = 0 , for al l v ∈ H 1 Γ (Ω; w ) . Pr o of. Let u ∈ [ C ∞ (Ω)] 2 . Then for all 0 < δ 1 ≤ δ 2 ≤ 1 8 , Z δ 2 δ 1 1 δ g ( δ ; u , v ) 2 d δ = Z δ 2 δ 1 1 δ ⟨ u , v ⟩ 2 H − 1 2 (Γ ∗ δ ) ,H 1 2 (Γ ∗ δ ) d δ = Z δ 2 δ 1 1 δ Z Γ ∗ δ ( u · ν ) v d S ! 2 d δ ≤ 2 π Z δ 2 δ 1 Z Γ ∗ δ 1 δ u 2 2 v 2 d S d δ ≤ 2 π ∥ v ∥ 2 L ∞ (Ω) Z T × ( δ 1 ,δ 2 ) r − 1 u 2 2 d z 14 for all v ∈ C ∞ Γ (Ω). F rom Lemma 2.12 w e get Z δ 2 δ 1 1 δ g ( δ ; u , v ) 2 d δ ≤ 2 π ∥ v ∥ 2 L ∞ (Ω) Z T × ( δ 1 ,δ 2 ) r − 1 u 2 2 d z ≤ 2 π ∥ v ∥ 2 L ∞ (Ω) Z T × ( δ 1 ,δ 2 ) u · A − 1 u d z for all u ∈ H (div , Ω) and all v ∈ C ∞ Γ (Ω), since r − 1 = r α − 1 r − α ≤ r − α . Hence, for all u ∈ H (div , Ω; w − 1 ) and all v ∈ C ∞ Γ (Ω), Z δ 2 δ 1 1 δ g ( δ ; u , v ) 2 d δ ≤ 2 π ∥ v ∥ 2 L ∞ (Ω) ∥ u ∥ H (div , Ω; w − 1 ) < + ∞ , this implies that g (0; u , v ) = 0 (2.5) for all u ∈ H (div , Ω; w − 1 ) and all v ∈ C ∞ Γ (Ω). Hence, from Lemma 2.11 , for an y u ∈ H (div , Ω; w − 1 ) and v ∈ C ∞ Γ (Ω), T w ( u , v ) = 0 . Finally , from Lemma 2.13 , w e get T w ( u , v ) = 0 , for all u ∈ H (div , Ω; w − 1 ) , v ∈ H 1 Γ (Ω; w ) . This completes the pro of of this theorem. Corollary 2.15. F or e ach u = ( u 1 , u 2 ) ∈ H (div , Ω; w − 1 ) , we have T w ( u , v ) = Z Γ v u 2 d s, for al l v ∈ H 1 (Ω; w ) . Pr o of. Cho osing ζ = ζ ( r ) ∈ C ∞ (Ω) , 0 ≤ ζ ≤ 1 such that ζ = 1 on  −∞ , 1 2  , ζ = 0 on  3 4 , + ∞  . Then v 1 = ζ v ∈ H 1 Γ (Ω; w ) and v 2 = (1 − ζ ) v ∈ H 1 0 ( T × ( 1 4 , 1)) and v = v 1 + v 2 . Then T w ( u , v ) = T w ( u , v 1 + v 2 ) = Z Ω  [div u ] v 1 + u · ∇ v 1  d z + Z Ω  [div u ] v 2 + u · ∇ v 2  d z = Z Ω  [div u ] v 2 + u · ∇ v 2  d z = Z Ω div( u v 2 )d z = Z Γ v 2 u · ν d S = Z Γ v u 2 d S. b y Remark 2.9 and Theorem 2.14 . This completes the pro of of the corollary . Lemma 2.16. The sp ac e C ∞ (Ω) ∩ H (div , Ω; w − 1 ) is dense in H (div , Ω; w − 1 ) . Pr o of. It is easily verified that [ C ∞ 0 (Ω)] 2 ⊆ H (div , Ω; w − 1 ). 15 Let h ∈ H (div , Ω; w − 1 ) b e orthogonal to [ C ∞ (Ω)] 2 ∩ H (div , Ω; w − 1 ). i.e., Z Ω h · A − 1 v d x + Z Ω [div h ] [div v ]d x = 0 , ∀ v ∈ [ C ∞ (Ω)] 2 ∩ H (div , Ω; w − 1 ) . Set g = div h , then Z Ω g div v d z = − Z Ω h · A − 1 v d z , ∀ v ∈ [ C ∞ (Ω)] 2 ∩ H (div , Ω; w − 1 ) . (2.6) Hence ∇ g = A − 1 h in the sense of distributions on Ω by [ C ∞ 0 (Ω)] 2 ⊆ H (div , Ω; w − 1 ). There- fore A ∇ g = h , and g ∈ H 1 (Ω; w ). Define e g ( z ) = ( g ( z ) , z ∈ Ω , 0 , z ∈ T × (1 , 2) , f ∇ g ( z ) = ( A − 1 ( z ) h ( z ) , z ∈ Ω , 0 , z ∈ T × (1 , 2) , then, for each ψ ∈ [ C ∞ 0 ( T × (0 , 2))] 2 , we ha ve ψ ∈ H (div , Ω; w − 1 ) by (supp ψ ) ∩ Γ ∗ = ∅ , and hence Z T × (0 , 2) f ∇ g · ψ d z = Z Ω h · A − 1 ψ d z = − Z Ω g div ψ d z = − Z T × (0 , 2) e g div ψ d z = Z T × (0 , 2) ∇ e g · ψ d z , where we used ( 2.6 ) in the second equalit y . Hence f ∇ g = ∇ e g on T × (0 , 2) and e g ∈ H 1 ( T × (0 , 2); w − 1 ). This shows that e g | Γ = 0 b y Remark 2.1 , and hence g ∈ H 1 Γ (Ω; w ). Finally , for eac h e h ∈ H (div , Ω; w − 1 ), we hav e ( h , e h ) H (div , Ω; w − 1 ) = Z Ω  h · A − 1 e h + (div h )(div e h )  d z = Z Ω  e h · ∇ g + (div e h ) g  d z = T w ( e h , g ) = 0 b y g ∈ H 1 Γ (Ω; w ) and Theorem 2.14 . Hence h = 0. This implies that C ∞ (Ω) ∩ H (div , Ω; w − 1 ) is dense in H (div , Ω; w − 1 ). This completes the pro of of the lemma. 2.3 Neumann b oundary condition The following theorem shows that the b oundary condition ∂ φ ∂ ν A = 0 on Γ ∗ × (0 , T ) arises naturally in the equation ( 1.1 ). This differs from the classical Neumann b oundary condition for uniformly hyperb olic equations. Theorem 2.17. The fol lowing assertions hold true: 16 (i) The fol lowing inclusion and e quality hold H (div , Ω; w − 1 ) ⊆ k er γ ν = { u ∈ H (div , Ω) : γ ν u = 0 } . (ii) Supp ose u ∈ H 2 (Ω; w ) . Then ∂ u ∂ ν A = γ ν ( A ∇ u ) = 0 on Γ ∗ . F urthermor e, for al l u ∈ H 2 (Ω; w ) and v ∈ H 1 Γ (Ω; w ) , we have − Z Ω [div( A ∇ u )] v d z = Z Ω ∇ v · A ∇ u d z . Pr o of. (i) Let u ∈ H (div , Ω; w − 1 ). F or all v ∈ C ∞ Γ (Ω), from Lemma 2.11 and Theorem 2.14 , w e hav e ⟨ γ ν u , v ⟩ H − 1 2 (Γ ∗ ) ,H 1 2 (Γ ∗ ) = T w ( u , v ) = 0 , and then γ ν u = 0 according to C ∞ (Γ ∗ ) ∼ = C ∞ ( T ) = C ∞ 0 ( T ) is dense in H 1 2 (Γ ∗ ) = H 1 2 ( T ). (ii) Since u ∈ H 2 (Ω; w ), w e get A ∇ u ∈ H (div , Ω; w − 1 ). F rom (i) w e obtain ∂ u ∂ ν A = γ ν ( A ∇ u ) = 0 on Γ ∗ . Finally , from u ∈ H 2 (Ω; w ) and v ∈ H 1 Γ (Ω; w ), w e hav e A ∇ u ∈ H (div , Ω; w − 1 ), and − Z Ω [div( A ∇ u )] v d z = Z Ω ∇ v · A ∇ u d z b y Theorem 2.14 . This completes the pro of of this theorem. Corollary 2.18. L et u ∈ H 2 (Ω; w ) and v ∈ H 1 (Ω; w ) . Then − Z Ω [div( A ∇ u )] v d z = Z Ω ∇ v · A ∇ u d z + Z Γ ∂ u ∂ ν A v d S, wher e ∂ u ∂ ν A = A ∇ u · ν on Γ . Pr o of. Cho osing ζ = ζ ( r ) ∈ C ∞ (Ω) , 0 ≤ ζ ≤ 1 such that ζ = 1 on  −∞ , 1 2  , ζ = 0 on  3 4 , + ∞  . Then v 1 = ζ v ∈ H 1 Γ (Ω; w ) and v 2 = (1 − ζ ) v ∈ H 1 0 ( T × ( 1 4 , 1)) and v = v 1 + v 2 . Hence − Z Ω [div( A ∇ u )] v d z = − Z Ω [div( A ∇ u )] v 1 d z − Z T × ( 1 4 , 1) [div( A ∇ u )] v 2 d z = Z Ω ∇ u · A ∇ v 1 d z + Z T × ( 1 4 , 1) A ∇ u · ∇ v 2 d z − Z Γ ∂ u ∂ ν A v 2 d S = Z Ω ∇ u · A ∇ v d z − Z Γ ∂ u ∂ ν A v d S. This completes the pro of of the corollary . 17 2.4 The sp ectrum for the partial differen tial op erator A After defining the Neumann b oundary condition, we in tro duce the partial differential op erator A . Associated with this operator are both degenerate elliptic and degenerate hy- p erb olic equations. In this subsection, we focus on the existence of weak solutions to the degenerate elliptic equation and on the sp ectral prop erties of the op erator A . More precisely , A u = − div( A ∇ u ) = − ∂ θθ u − ∂ r ( r α ∂ r u ) . W e consider the follo wing degenerate elliptic equation        A u = g , in Ω , u = 0 , on Γ , ∂ u ∂ ν A = 0 , on Γ ∗ , (2.7) where Ω , Γ , Γ ∗ are defined in Assumption 1.1 , and g ∈ L 2 (Ω). W e call u ∈ H 1 Γ (Ω; w ) ∩ H 2 (Ω; w ) a weak solution of ( 2.7 ) if Z Ω ∇ u · A ∇ v d z = Z Ω g v d z , for all v ∈ H 1 Γ (Ω; w ) . Lemma 2.19. The e quation ( 2.7 ) has a unique we ak solution u ∈ H 1 Γ (Ω; w ) ∩ H 2 (Ω; w ) . Mor e over, ther e exists a p ositive c onstant C , dep ending only on α , such that ∥ u ∥ H 1 Γ (Ω; w ) ≤ C ∥ g ∥ L 2 (Ω) . (2.8) Pr o of. Define B [ u, v ] = Z Ω ∇ u · A ∇ v d z , then, from Remark 2.4 , for all u, v ∈ H 1 Γ (Ω; w ), w e hav e ∥ u ∥ 2 H 1 Γ (Ω; w ) = B [ u, u ] , | B [ u, v ] | ≤ ∥ u ∥ H 1 Γ (Ω; w ) ∥ v ∥ H 1 Γ (Ω; w ) . Note that g ∈ L 2 (Ω) ⊆ H − 1 (Ω; w ) since Z Ω g v d z ≤ ∥ g ∥ L 2 (Ω) ∥ v ∥ L 2 (Ω) ≤ C ∥ g ∥ L 2 (Ω) ∥ v ∥ H 1 Γ (Ω; w ) b y Remark 2.4 , hence, from the Lax-Milgram theorem, there exists a weak unique u ∈ H 1 Γ (Ω; w ) to the equation ( 2.7 ). Moreo ver, from ∥ u ∥ 2 H 1 Γ (Ω; w ) ≤ C ∥ g ∥ L 2 (Ω) ∥ u ∥ H 1 Γ (Ω; w ) w e get ( 2.8 ). Finally , from A u = g in the sense of distributions, we get u ∈ H 2 (Ω; w ). W e complete the pro of of this lemma. 18 The follo wing theorem pro vides an impro ved regularity result for solutions of ( 2.7 ). It will b e used later in the analysis of the degenerate hyperb olic equation ( 1.1 ). Theorem 2.20. L et g ∈ L 2 (Ω) . L et u b e the solution of ( 2.7 ) with r esp e ct to g . Then ∂ θθ u ∈ L 2 (Ω) , and r α 2 ∂ θr u ∈ L 2 (Ω) , and r 1+ α 2 ∂ rr u ∈ L 2 (Ω) . Mor e over, ther e exists a p ositive c onstant C , dep ending only on α , such that Z Ω  ( ∂ θθ u ) 2 + r α ( ∂ θr u ) 2 + r 2+ α ( ∂ rr u ) 2  d z ≤ C Z Ω g 2 d z . (2.9) Pr o of. W e denote the difference quotien t D h θ u = 1 h  u ( θ + h, r ) − u ( θ , r )  . F or eac h θ 0 ∈ T , choosing δ 0 ∈ (0 , 1 16 ) and ζ = ζ ( θ ) ∈ C ∞ ( T ) , 0 ≤ ζ ≤ 1 such that ζ = 0 on ( θ 0 − δ 0 , θ 0 + δ 0 ) , ζ = 1 on T − ( θ 0 − 2 δ 0 , θ 0 + 2 δ 0 ) , and | ζ ′ | ≤ C and | ζ ′′ | ≤ C , where the constants C > 0 are absolute. T aking v = − D − h θ  ζ 2 D h θ u  for h ∈ (0 , δ 0 ) , then v ∈ H 1 Γ (Ω; w ). Using v as the test function, from ( 2.7 ), w e get A 1 ≡ Z Ω ∇ u · A ∇ v d z = Z Ω g v d z ≡ A 2 . Note that for the functions u, v on Ω, we ha v e D h θ ∇ v = ∇ D h θ v , Z Ω v D − h θ u d z = − Z Ω uD h θ v d z , then A 1 = Z Ω ∇ u · A ∇  − D h θ ( ζ 2 D h θ u )  d z = Z Ω ( ∂ θ u ) ∂ θ  − D h θ ( ζ 2 D h θ u )  d z + Z Ω r α ( ∂ r u ) ∂ r  − D h θ ( ζ 2 D h θ u )  d z , 19 i.e., A 1 = Z Ω ( D h θ D θ u ) ∂ θ ( ζ 2 D h θ u )d z + Z Ω r α ( D h θ ∂ r u ) ∂ r ( ζ 2 D h θ u )d z = Z Ω ζ 2  ( D h θ ∂ θ u ) 2 + r α ( D h θ ∂ r u ) 2  d z + 2 Z Ω ζ ( ∂ θ ζ )( D h θ ∂ θ u ) D h θ u d z ≥ 1 2 Z Ω ζ 2  ( D h θ ∂ θ u ) 2 + r α ( D h θ ∂ r u ) 2  d z − 4 Z Ω ( ∂ θ ζ ) 2 ( D h θ u ) 2 d z . This, together with [ 8 , Theorem 3 (i) (p. 292) in Chapter 5.8.2], we get A 1 ≥ 1 2 Z Ω ζ 2  ( D h θ ∂ θ u ) 2 + r α ( D h θ ∂ r u ) 2  d z − C Z Ω | ∂ θ u | 2 d z , where the constant C > 0 is absolute. No w, from [ 8 , Theorem 3 (i) (p. 292) in Chapter 5.8.2], we get A 2 ≤ 1 2 Z Ω g 2 d z + 1 2 Z Ω v 2 d z ≤ 1 2 Z Ω g 2 d z + C Z Ω   ∂ θ  ζ 2 D h θ u    2 d z ≤ 1 2 Z Ω g 2 d z + C Z Ω | ∂ θ u | 2 d z + 1 4 Z Ω ζ 2  D h θ ∂ θ u  2 d z . Then Z Ω ζ 2  ( D h θ ∂ θ u ) 2 + r α ( D h θ ∂ r u ) 2  d z ≤ C Z Ω | ∂ θ u | 2 d z + C Z Ω g 2 d z , and hence Z Ω ζ 2  ( ∂ θθ u ) 2 + r α ( ∂ θr u ) 2  d z ≤ C Z Ω g 2 d z , (2.10) b y ( 2.8 ) and [ 8 , Theorem 3 (ii) (p. 292) in Chapter 5.8.2], where the constan t C > 0 dep ending only on α . This implies ∂ θθ u ∈ L 2 (Ω) and r α 2 ∂ θr u ∈ L 2 (Ω) . Finally , from ( 2.10 ) and − ∂ θθ u − ∂ r ( r α ∂ r u ) = g w e get ( 2.9 ) and ∂ r ( r α ∂ r u ) ∈ L 2 (Ω) and r 1+ α 2 ∂ rr u ∈ L 2 (Ω) , b y r 1 − α 2 ∂ r ( r α ∂ r u ) = αr α 2 ∂ r u + r 1+ α 2 ∂ rr u. This completes the pro of of this theorem. 20 The follo wing lemma will also b e used in the analysis of the degenerate hyperb olic equa- tion ( 1.1 ). Lemma 2.21. L et u b e the we ak solution of ( 2.7 ) with r esp e ct to g . Then we have (i) ∂ θ u = 0 on Γ , (ii) for any  ∈ [0 , 2 − α 4 ) , r α 2 u 2 ∈ W 1 , 1 (Ω) , and r α 2 + ϵ u ∈ H 1 (Ω) , and r u 2 = r u = 0 on Γ ∗ , (iii) and r ( ∇ u · A ∇ u ) ∈ W 1 , 1 (Ω) , and r ( ∇ u · A ∇ u ) = 0 on Γ ∗ . Pr o of. W e prov e this lemma by the follo wing steps. Step 1 . W e prov e (i). Since u = 0 on Γ = T × { 1 } , then ∂ θ u = 0 on Γ. This prov es (i). Step 2 . W e prov e (ii). Let u ∈ H 1 Γ (Ω; w ). F rom R Ω r α 2 u 2 d z ≤ R Ω u 2 d z , and from Lemma 2.3 we obtain Z Ω   ∇ ( r α 2 u 2 )   d z ≤ C Z Ω r α 2 − 1 u 2 d z + C Z Ω ∇ u · A ∇ u d z ≤ C Z Ω ∇ u · A ∇ u d z , where the constan ts C > 0 depending only α , hence r α 2 u 2 ∈ W 1 , 1 (Ω) and ru 2 = r 1 − α 2 ( r α 2 u 2 ) = 0 on Γ ∗ . Since R Ω ( r α 2 + ϵ u ) 2 d z ≤ R Ω u 2 d z with  ∈ [0 , 2 − α 4 ), and from Lemma 2.3 we get Z Ω   ∇ ( r α 2 + ϵ u )   2 d z ≤ C Z Ω ( ∂ θ u ) 2 d z + C Z Ω r α ( ∂ r u ) 2 d z + C Z Ω u 2 r α +2 ϵ − 2 d z ≤ C Z Ω ∇ u · A ∇ u d z , w e obtain r α 2 + ϵ u ∈ H 1 (Ω). Hence r u = r 1 − α 2 − ϵ ( r α 2 + ϵ u ) = 0 on Γ ∗ . Since R Ω r ( ∂ θ u ) 2 d z ≤ R Ω ( ∂ θ u ) 2 d z , and Z Ω   ∂ θ  r ( ∂ θ u ) 2    d z ≤ Z Ω ( ∂ θ u ) 2 d z + Z Ω ( ∂ θθ u ) 2 d z , and Z Ω   ∂ r  r ( ∂ θ u ) 2    d z ≤ 2 Z Ω ( ∂ θ u ) 2 d z + Z Ω r α ( ∂ θr u ) 2 d z , these imply r ( ∂ θ u ) 2 ∈ W 1 , 1 (Ω) by Theorem 2.20 . Step 3 . W e prov e (iii). Since R Ω r 1+ α ( ∂ r u ) 2 d z ≤ R Ω r α ( ∂ r u ) 2 d z , and Z Ω   ∂ θ  r 1+ α ( ∂ r u ) 2    d z ≤ Z Ω r α ( ∂ r u ) 2 d z + Z Ω r α ( ∂ θr u ) 2 d z , 21 and Z Ω   ∂ r  r 1+ α ( ∂ r u ) 2    d z ≤ (2 + α ) Z Ω r α ( ∂ r u ) 2 d z + Z Ω r 2+ α ( ∂ rr u ) 2 d z . This implies that r 1+ α ( ∂ r u ) 2 ∈ W 1 , 1 (Ω). F rom ab ov e, we get r ( ∇ u · A ∇ u ) ∈ W 1 , 1 (Ω). Since ∂ θ u ∈ L 2 (Ω), and from Theorem 2.20 we kno w ∂ θ ( ∂ θ u ) = ∂ θθ u ∈ L 2 (Ω) , and r α 2 ∂ r ( ∂ θ u ) = r α 2 ∂ θr u ∈ L 2 (Ω) , and from (i) w e get ∂ θ u ∈ H 1 Γ (Ω; w ). Hence r ( ∂ θ u ) 2 = 0 on Γ ∗ b y (ii) (replace u by ∂ θ u in (ii)). Finally , w e pro ve r 1+ α ( ∂ r u ) 2 = 0 on Γ ∗ . Since R Ω ( r ∂ r u ) 2 d z ≤ R Ω r α ( ∂ r u ) 2 d z , and Z Ω ( ∂ θ [ r ∂ r u ]) 2 d z ≤ Z Ω r α ( ∂ θr u ) 2 d z , and Z Ω r α ( ∂ r [ r ∂ r u ]) 2 d z ≤ 2 Z Ω r α ( ∂ r u ) 2 d z + 2 Z Ω r 2+ α ( ∂ rr u ) 2 d z , then r ∂ r u ∈ H 1 (Ω; w ). T aking ζ = ζ ( r ) ∈ C ∞ ( R ) , 0 ≤ ζ ≤ 1 such that ζ = 1 on  0 , 1 2  , ζ = 0 on  3 4 , 1  , | ζ ′ | ≤ C, where the constant C > 0 is absolute. Hence we get − Z Γ ∗ δ r 1+ α ( ∂ r u ) 2 d S = g ( δ ; A ∇ u, ζ r ∂ r u ) = ⟨ γ ν ( A ∇ u ) , ζ r ∂ r u ⟩ H − 1 2 (Γ ∗ δ ) ,H 1 2 (Γ ∗ δ ) → 0 as δ → 0 + b y u ∈ H 2 (Ω; w ) and Lemma 2.12 and Theorem 2.14 and ( 2.5 ). Therefore, r 1+ α ( ∂ r u ) 2 = 0 on Γ ∗ . W e complete the pro of of this lemma. 2.5 Sp ectrum Notation 2.22. Now, from Lemma 2.5 , the partial differential op erator A enjo ys a discrete sp ectrum 0 < λ 1 < λ 2 ≤ λ 3 ≤ · · · → + ∞ . 22 i.e., λ n ( n ∈ N ∗ ) is the solution of the following equation        A Φ n = λ n Φ n , in Ω , Φ n = 0 , on Γ , ∂ Φ n ∂ ν A = 0 , on Γ ∗ , (2.11) and Φ n ∈ D ( A ) is called the eigenfunction of A with resp ect to the eigen v alue λ n . W e denote Φ n ( n ∈ N ∗ ) is the orthonormal basis of L 2 (Ω). See [ 8 , Theorem 7 in App endix D.6, p. 728]. Lemma 2.23. L et u = P ∞ i =1 u i Φ i ∈ H 1 Γ (Ω; w ) with u i = ( u, Φ i ) L 2 (Ω) for al l i ∈ N ∗ . We have ∇ u = P ∞ i =1 u i ∇ Φ i and ∥ u ∥ H 1 Γ (Ω; w ) = ( P ∞ i =1 u 2 i λ i ) 1 2 , and u ∈ H 2 (Ω; w ) ⇔ ∞ X i =1 u 2 i λ 2 i < ∞ , and A u = ∞ X i =1 u i λ i Φ i , and ∥A u ∥ L 2 (Ω) = ∞ X i =1 u 2 i λ 2 i ! 1 2 . Pr o of. F rom ( 2.11 ), from Φ n ∈ D ( A ) = H 2 (Ω; w ) ∩ H 1 Γ (Ω; w ) and Theorem 2.17 (ii), we ha v e Z Ω ∇ Φ k · A ∇ Φ l d z = δ kl λ k for k , l ∈ N ∗ , (2.12) where δ kl is the Kroneck er delta function, i.e., δ kl = 1 for k = l and δ kl = 0 for k  = l . W e now demonstrate that { λ − 1 2 k Φ k } ∞ k =1 forms an orthonormal basis of H 1 Γ (Ω; w ). F rom ( 2.12 ), it is straightforw ard to verify that { λ − 1 2 k Φ k } ∞ k =1 is an orthonormal subset of H 1 Γ (Ω; w ). T o prov e it is a basis, assume by con tradiction that there exists 0  = u ∈ H 1 Γ (Ω; w ) suc h that ( u, Φ k ) H 1 Γ (Ω; w ) = Z Ω ∇ Φ k · A ∇ u d z = 0 for all k ∈ N ∗ . Giv en that { Φ k } k ∈ N is an orthonormal basis of L 2 (Ω), for u ∈ H 1 Γ (Ω; w ), w e can express u = ∞ X k =1 d k Φ k , where d k = ( u, Φ k ) L 2 (Ω) , k ∈ N . (2.13) Then, from ( 2.11 ), we hav e 0 = ( u, Φ k ) H 1 Γ (Ω; w ) = Z Ω ∇ Φ k · A ∇ u d z = λ k Z Ω Φ k u d z = λ k d k , whic h implies d k = 0. This leads to u = 0, a contradiction. 23 F rom ab ov e, since u ∈ H 1 Γ (Ω; w ), w e hav e u = ∞ X i =1 e i  λ − 1 2 i Φ k  in H 1 Γ (Ω; w ) , and ∥ u ∥ 2 H 1 Γ (Ω; w ) = ∞ X i =1 e 2 i , and e i = Z Ω ∇ u · A  λ − 1 2 i ∇ Φ i  d z = λ − 1 2 i Z Ω ∇ u · A ∇ Φ i d z = λ 1 2 i Z Ω u Φ i d z = λ 1 2 i u i , hence, ∇ u = P ∞ i =1 u i ∇ Φ i . Moreov er, ∥ u ∥ H 1 Γ (Ω; w ) = ( P ∞ i =1 λ i u 2 i ) 1 2 . Let u ∈ H 2 (Ω; w ). T ak e ϕ n = P n i =1 u i Φ k for each n ∈ N ∗ , then u ∈ D ( A ), and from Φ i ∈ D ( A ) ( i ∈ N ∗ ) and Theorem 2.17 (ii), we get ( A u, A ϕ n ) L 2 (Ω) = n X i =1 u i λ i Z Ω ( A u )Φ i d z = n X i =1 u i λ i Z Ω ∇ u · A ∇ ϕ d z = n X i =1 u i λ i Z Ω u A Φ i d z = n X i =1 u i λ 2 i Z Ω u Φ i d z = n X i =1 u 2 i λ 2 i and ∥A ϕ n ∥ 2 L 2 (Ω) = P n i =1 u 2 i λ 2 i w e obtain n X i =1 u 2 i λ 2 i ≤ ∥A u ∥ 2 L 2 (Ω) for all n ∈ N by Cauc hy inequality . This implies that P ∞ i =1 u 2 i λ 2 i ≤ ∥A u ∥ 2 L 2 (Ω) < ∞ . Let P ∞ i =1 u 2 i λ 2 i < ∞ . F or eac h ϕ ∈ C ∞ 0 (Ω), we hav e ( A u, ϕ ) L 2 (Ω) = Z Ω ∇ u · A ∇ ϕ d z = ∞ X i =1 u i Z Ω ∇ Φ i · A ∇ ϕ d z = ∞ X i =1 u i λ i Z Ω Φ i ϕ d z in the sense of distributions. Note that Z Ω n X i =1 u i λ i Φ i ! 2 d z = n X i =1 u 2 i λ 2 i ≤ ∞ X i =1 u 2 i λ 2 i < ∞ for all n ∈ N , i.e., P ∞ i =1 u i λ i Φ i ∈ L 2 (Ω). Hence ( A u, ϕ ) L 2 (Ω) = ∞ X i =1 u i λ i Φ i , ϕ ! L 2 (Ω) . 24 This implies that A u = P ∞ i =1 u i λ i Φ i ∈ L 2 (Ω), and hence ∥A u ∥ L 2 (Ω) ≤ ∞ X i =1 u 2 i λ 2 i ! 1 2 . W e complete the pro of of this lemma. 2.6 Existence and uniqueness of w eak solutions to equation ( 1.1 ) In this subsection, w e establish the existence and uniqueness of w eak solutions to equation ( 1.1 ). Definition 2.24. Let ϕ 0 ∈ H 1 Γ (Ω; w ) , ϕ 1 ∈ L 2 (Ω) and f ∈ L 2 ( Q ). A function ϕ ∈ L 2 (0 , T ; H 1 Γ (Ω; w )) ∩ H 1 (0 , T ; L 2 (Ω)) ∩ H 2 (0 , T ; H − 1 Γ (Ω; w )) is a weak solution of the equation ( 1.1 ) with resp ect to ( ϕ 0 , ϕ 1 , f ), provided (i) for each v ∈ H 1 Γ (Ω; w ) and a.e. t ∈ [0 , T ], we ha ve ⟨ ∂ tt ϕ, v ⟩ H − 1 Γ (Ω; w ) ,H 1 Γ (Ω; w ) + ( ϕ, v ) H 1 Γ (Ω; w ) = ( f , v ) L 2 (Ω) , (ii) ϕ (0) = ϕ 0 and ∂ t ϕ (0) = ϕ 1 . The following lemma is [ 4 , Lemma 2.3 (p. 61)]. Lemma 2.25. L et ( M , g ) b e a C m -R iemannian manifold with c omp act b oundary ∂ M . Then ther e exists a C m − 1 -ve ctor field n such that n ( x ) = ν ( x ) , x ∈ ∂ M , and | n ( x ) | ≤ 1 , x ∈ M , wher e ν is the unit outwar d normal ve ctor to ∂ M . W e are no w in a p osition to establish the existence of weak solutions to equation ( 1.1 ). Theorem 2.26. Under Assumption 1.1 , the e quation ( 1.1 ) with r esp e ct to ( ϕ 0 , ϕ 1 , f ) has a unique we ak solution ϕ ∈ L 2 (0 , T ; H 1 Γ (Ω; w )) ∩ H 1 (0 , T ; L 2 (Ω)) ∩ H 2 (0 , T ; H − 1 Γ (Ω; w )) satisfies the fol lowing estimate ess sup t ∈ [0 ,T ]  ∥ ϕ ( t ) ∥ H 1 Γ (Ω; w ) + ∥ ∂ t ϕ ( t ) ∥ L 2 (Ω)  + ∥ ∂ tt ϕ ∥ L 2 (0 ,T ; H − 1 Γ (Ω; w )) +     ∂ ϕ ∂ ν     L 2 (0 ,T ; L 2 (Γ)) ≤ C  ∥ ϕ 0 ∥ H 1 Γ (Ω; w ) + ∥ ϕ 1 ∥ L 2 (Ω) + ∥ f ∥ L 2 ( Q )  , (2.14) 25 wher e the c onstant C > 0 dep ending only on α and T . Mor e over, if we further assume ϕ 0 ∈ D ( A ) , ϕ 1 ∈ H 1 Γ (Ω; w ) and f ∈ H 1 (0 , T ; L 2 (Ω)) , then the we ak solution of ( 1.1 ) with r esp e ct to ( ϕ 0 , ϕ 1 , f ) satisfy ϕ ∈ L 2 (0 , T ; D ( A )) ∩ H 1 (0 , T ; H 1 Γ (Ω; w )) ∩ H 2 (0 , T ; L 2 (Ω)) , and we have the estimate ess sup t ∈ [0 ,T ]  ∥ ϕ ( t ) ∥ H 2 (Ω 1 2 ) + ∥ ϕ ( t ) ∥ D ( A ) + ∥ ∂ t ϕ ( t ) ∥ H 1 Γ (Ω; w ) + ∥ ∂ tt ϕ ( t ) ∥ L 2 (Ω)  +     ∂ ϕ ∂ ν     H 1 (0 ,T ; L 2 (Γ)) ≤ C  ∥ ϕ 0 ∥ D ( A ) + ∥ ϕ 1 ∥ H 1 Γ (Ω; w ) + ∥ f ∥ H 1 (0 ,T ; L 2 (Ω))  , (2.15) wher e the c onstant C > 0 dep ending only on α and T . Pr o of. W e prov e this theorem by the follo wing steps. Step 1 . Galerkin’s metho d. Let 2 ≤ k ∈ N ∗ . Define ϕ k = k X n =1 ϕ k n ( t )Φ n ( x ) , and f k = k X n =1 f k n ( t )Φ n ( x ) . (2.16) where ϕ k n ( t ) , n = 1 , · · · , k is the solution of the following system d 2 d t 2 ϕ k n ( t ) + λ n ϕ k n ( t ) = f k n ( t ) , n = 1 , · · · , k (2.17) with ϕ k n (0) = ( ϕ 0 , Φ n ) L 2 (Ω) , d ϕ k n d t (0) = ( ϕ 1 , Φ n ) L 2 (Ω) , f k n = ( f , Φ n ) L 2 (Ω) , n = 1 , · · · , k . (2.18) It is well-kno wn that the system ( 2.17 ) and ( 2.18 ) has a unique solution ( ϕ k 1 , · · · , ϕ k k ) for t ∈ [0 , T ]. W e note that ( 2.17 ) is equiv alen t to the following equality ( ∂ tt ϕ k , Φ n ) L 2 (Ω) + ( ϕ k , Φ n ) H 1 Γ (Ω; w ) = ( f k , Φ n ) L 2 (Ω) , n = 1 , · · · , k . (2.19) Step 2 . Energy estimate. Multiplying ( 2.19 ) by d d t ϕ k n ( t ), summing n = 1 , · · · , k , then ( ∂ tt ϕ k , ∂ t ϕ k ) L 2 (Ω) + ( ϕ k , ∂ t ϕ k ) H 1 Γ (Ω; w ) = ( f k , ∂ t ϕ k ) L 2 (Ω) , for a.e. t ∈ [0 , T ] . 26 Note that ( ∂ tt ϕ k , ∂ t ϕ k ) L 2 (Ω) = 1 2 d d t ∥ ∂ t ϕ k ∥ 2 L 2 (Ω) , ( ϕ k , ∂ t ϕ k ) H 1 Γ (Ω; w ) = 1 2 d d t ∥ ϕ k ∥ 2 H 1 Γ (Ω; w ) , w e get d d t  ∥ ∂ t ϕ k ∥ 2 L 2 (Ω) + ∥ ϕ k ∥ 2 H 1 Γ (Ω; w )  ≤ ∥ f k ∥ 2 L 2 (Ω) + ∥ ∂ t ϕ k ∥ 2 L 2 (Ω) , and then ∥ ∂ t ϕ k ∥ 2 L 2 (Ω) + ∥ ϕ k ∥ 2 H 1 Γ (Ω; w ) ≤ e t  Z t 0 ∥ f k ∥ 2 L 2 (Ω) d t + ∥ ∂ t ϕ k (0) ∥ 2 L 2 (Ω) + ∥ ϕ k (0) ∥ 2 H 1 Γ (Ω; w )  ≤ e T  ∥ f ∥ 2 L 2 ( Q ) + ∥ ϕ 1 ∥ 2 L 2 (Ω) + ∥ ϕ 0 ∥ 2 H 1 Γ (Ω; w )  (2.20) for all t ∈ [0 , T ] by the Gron wall’s inequality and Lemma 2.23 . F or each v ∈ H 1 Γ (Ω; w ) with ∥ v ∥ H 1 Γ (Ω; w ) ≤ 1, w e write v = v 1 + v 2 with v 1 ∈ span { Φ n } k n =1 and ( v 2 , Φ n ) L 2 (Ω) = 0 for all n = 1 , · · · , k , then, from ( 2.16 ) and ( 2.19 ), w e get  ∂ tt ϕ k , v  H − 1 Γ (Ω; w ) ,H 1 Γ (Ω; w ) = ( ∂ tt ϕ k , v 1 ) L 2 (Ω) = ( f k , v 1 ) L 2 (Ω) − ( ϕ k , v 1 ) H 1 Γ (Ω; w ) . Whic h together with ( 2.20 ) we obtain ∥ ∂ tt ϕ k ∥ H − 1 Γ (Ω; w ) ≤ C  ∥ f k ∥ L 2 (Ω) + ∥ ϕ k ∥ H 1 Γ (Ω; w )  b y Remark 2.4 and ∥ v 1 ∥ H 1 Γ (Ω; w ) ≤ 1, where the constant C > 0 dep ends only on α . Hence, from ( 2.20 ), we get ∥ ∂ tt ϕ k ∥ L 2 (0 ,T ; H − 1 Γ (Ω; w )) ≤ C  ∥ f ∥ L 2 ( Q ) + ∥ ϕ 1 ∥ L 2 (Ω) + ∥ ϕ 0 ∥ H 1 Γ (Ω; w )  , (2.21) where the constant C > 0 dep ends only on α and T . Step 3 . Approximation, or w eak conv ergence. F rom ( 2.20 ) and ( 2.21 ), there exists ψ ∈ L 2 (0 , T ; H 1 Γ (Ω; w )) ∩ H 1 (0 , T ; L 2 (Ω)) ∩ H 2 (0 , T ; H − 1 Γ (Ω; w )) suc h that ϕ k → ψ w eak star in L ∞ (0 , T ; H 1 Γ (Ω; w )) , ∂ t ϕ k → ∂ t ψ w eak star in L ∞ (0 , T ; L 2 (Ω)) , ∂ tt ϕ k → ∂ tt ψ w eakly in L 2 (0 , T ; H − 1 Γ (Ω; w )) (2.22) 27 b y abstract subsequence. Hence ess sup t ∈ [0 ,T ]  ∥ ∂ t ψ ∥ L 2 (Ω) + ∥ ψ ∥ H 1 Γ (Ω; w )  + ∥ ∂ tt ψ ∥ L 2 (0 ,T ; H − 1 Γ (Ω; w )) ≤ C  ∥ f ∥ L 2 ( Q ) + ∥ ϕ 1 ∥ L 2 (Ω) + ∥ ϕ 0 ∥ H 1 Γ (Ω; w )  , (2.23) where the constant C > 0 dep ends only on α and T . Moreov er, we hav e ψ ∈ C ([0 , T ]; L 2 (Ω)) , and ∂ t ψ ∈ C ([0 , T ]; H − 1 Γ (Ω; w )) . (2.24) Step 4 . Existence of weak solution to equation ( 1.1 ). W e show ψ is a solution of ( 1.1 ) with resp ect to ( ϕ 0 , ϕ 1 , f ). Let 2 ≤ l ∈ N ∗ . T aking h ∈ C 1 ([0 , T ]; H 1 Γ (Ω; w )) as the form h = l X n =1 h n ( t )Φ n ( x ) , with h n is a smo oth function on [0 , T ] , (2.25) m ultiplying h n ( t ) on the both sides of ( 2.19 ), summing n = 1 , · · · , l , integrating on [0 , T ], then, for all k ≥ l , we ha v e Z T 0  ∂ tt ϕ k , h  H − 1 Γ (Ω; w ) ,H 1 Γ (Ω; w ) d t + Z T 0 ( ϕ k , h ) H 1 Γ (Ω; w ) d t = Z T 0 ( f k , h ) L 2 (Ω) d t. (2.26) Whic h together with ( 2.22 ) we get Z T 0 ⟨ ∂ tt ψ , h ⟩ H − 1 Γ (Ω; w ) ,H 1 Γ (Ω; w ) d t + Z T 0 ( ψ , h ) H 1 Γ (Ω; w ) d t = Z T 0 ( f , h ) L 2 (Ω) d t. (2.27) Note that the set of the functions h defined in ( 2.25 ) is dense in L 2 (0 , T ; H 1 Γ (Ω; w )), hence ( 2.27 ) also hold for all h ∈ L 2 (0 , T ; H 1 Γ (Ω; w )). F or eac h t ∈ (0 , T ), for each  ∈ 1 4 min { t, T − t } , choose ζ δ ∈ C ∞ 0 (0 , T ) (0 < δ <  ) such that ζ δ = 1 on ( t − , t +  ) , ζ δ = 0 on ( t −  − δ , t +  + δ ) , then, for each v ∈ H 1 Γ (Ω; w ), w e hav e ζ δ v ∈ L 2 (0 , T ; H 1 Γ (Ω; w )) and Z T 0 ζ δ ⟨ ∂ tt ψ , v ⟩ H − 1 Γ (Ω; w ) ,H 1 Γ (Ω; w ) d t + Z T 0 ζ δ ( ψ , v ) H 1 Γ (Ω; w ) d t = Z T 0 ζ δ ( f , v ) L 2 (Ω) d t. b y ( 2.27 ). Letting δ → 0, we get Z t + ϵ t − ϵ ⟨ ∂ tt ψ , v ⟩ H − 1 Γ (Ω; w ) ,H 1 Γ (Ω; w ) d t + Z t + ϵ t − ϵ ( ψ , v ) H 1 Γ (Ω; w ) d t = Z t + ϵ t − ϵ ( f , v ) L 2 (Ω) d t. 28 F rom the Leb esgue p oint theorem, letting  → 0, we get ⟨ ∂ tt ψ , v ⟩ H − 1 Γ (Ω; w ) ,H 1 Γ (Ω; w ) + ( ψ , v ) H 1 Γ (Ω; w ) = ( f , v ) L 2 (Ω) for a.e. t ∈ [0 , T ] . This prov es Definition 2.24 (i). Next, we verify ψ (0) = ϕ 0 and ∂ t ψ (0) = ϕ 1 . F or any h ∈ C 2 ([0 , T ]; H 1 Γ (Ω; w )) with h ( T ) = ∂ t h ( T ) = 0, from ( 2.27 ) and ( 2.24 ), w e ha ve Z Z Q ψ ∂ tt h d z d t + Z Z Q ∇ ψ · A ∇ h d z d t = Z Z Q f h d z d t − ( ψ (0) , ∂ t h (0)) L 2 (Ω) + ⟨ ∂ t ψ (0) , h (0) ⟩ H − 1 Γ (Ω; w ) ,H 1 Γ (Ω; w ) . (2.28) Similarly from ( 2.26 ) we get Z Z Q ϕ k ∂ tt h d z d t + Z Z Q ∇ ϕ k · A ∇ h d z d t = Z Z Q f h d z d t − ( ϕ k (0) , ∂ t h (0)) L 2 (Ω) + ( ∂ t ϕ k , h (0)) L 2 (Ω) , this together with ( 2.18 ) and ( 2.22 ) we obtain Z Z Q ψ ∂ tt h d z d t + Z Z Q ∇ ψ · A ∇ h d z d t = Z Z Q f h d z d t − ( ϕ 0 , ∂ t h (0)) L 2 (Ω) + ( ϕ 1 , h (0)) L 2 (Ω) . Whic h together with ( 2.28 ) we deduce ψ (0) = ϕ 0 , and ∂ t ψ (0) = ϕ 1 . This prov e Definition ( 2.24 ) (ii). Step 5 . Uniqueness. W e only need to sho w that the w eak solution to equation ( 1.1 ) with resp ect to (0 , 0 , 0) is zero function. Let ϕ b e the solution of ( 1.1 ) with respect to ( ϕ 0 , ϕ 1 , f ) = (0 , 0 , 0). Fix s ∈ [0 , T ], taking h ( t ) = ( R s t ϕ ( τ )d τ , t ∈ [0 , s ] , 0 , t ∈ [ s, T ] , then h ( t ) ∈ H 1 Γ (Ω; w ) for a.e. t ∈ [0 , T ], and from Definition 2.24 (i) we hav e Z s 0 ⟨ ∂ tt ϕ, h ⟩ H − 1 Γ (Ω; w ) ,H 1 Γ (Ω; w ) d t + Z s 0 w ∇ ϕ · ∇ h d t = 0 . 29 Note that ∂ t ϕ (0) = h ( s ) = 0, integrating b y parts with resp ect to t , we get Z s 0 ( ∂ t ϕ, ϕ ) L 2 (Ω) d t − Z s 0 ( ∂ t h, h ) H 1 Γ (Ω; w ) d t = 0 b y ∂ t h = − ϕ (0 ≤ t ≤ s ), hence, 1 2 Z s 0 d d t ∥ ϕ ( t ) ∥ 2 L 2 (Ω) d t − 1 2 Z s 0 d d t ∥ h ∥ 2 H 1 Γ (Ω; w ) d t = 0 . Therefore, from ϕ (0) = 0 and h ( s ) = 0, we get ∥ ϕ ( s ) ∥ 2 L 2 (Ω) + ∥ h (0) ∥ 2 H 1 Γ (Ω; w ) = ∥ ϕ (0) ∥ 2 L 2 (Ω) + ∥ h ( s ) ∥ 2 H 1 Γ (Ω; w ) = 0 . i.e., ϕ ( s ) = 0. This shows that ϕ = 0 b y the arbitrary of s ∈ [0 , T ]. F rom the uniqueness and ( 2.23 ) we get the 1th-3th estimate in ( 2.14 ). Step 6 . Improv ed regularit y . Since f ∈ H 1 (0 , T ; L 2 (Ω)), from ( 2.17 ) we get d 2 d t 2  d ϕ k n d t  + λ n d ϕ k n d t ( t ) = d f k n d t ( t ) , n = 1 , · · · , k . Denote e ϕ k = ∂ t ϕ k , from ( 2.19 ), we hav e ( ∂ tt e ϕ k , Φ n ) L 2 (Ω) + ( e ϕ k , Φ n ) H 1 Γ (Ω; w ) = ( ∂ t f k , Φ n ) L 2 (Ω) , n = 1 , · · · , k . Multiplying d 2 φ k n d t 2 ( t ), summing n = 1 , · · · , k , w e discov er ( ∂ tt e ϕ k , ∂ t e ϕ k ) L 2 (Ω) + ( e ϕ k , ∂ t e ϕ k ) H 1 Γ (Ω; w ) = ( ∂ t f k , ∂ t e ϕ k n ) L 2 (Ω) . Similar to ( 2.20 ) in Step 2, we obtain ess sup t ∈ [0 ,T ]  ∥ ∂ t e ϕ k ∥ 2 L 2 (Ω) + ∥ e ϕ k ∥ 2 H 1 Γ (Ω; w )  ≤ e T  ∥ ∂ t f k ∥ 2 L 2 ( Q ) + ∥ ∂ t e ϕ k (0) ∥ 2 L 2 (Ω) + ∥ e ϕ k (0) ∥ 2 H 1 Γ (Ω; w )  . Note that from ( 2.18 ) we hav e e ϕ k (0) = ∂ t ϕ k (0) = P k n =1 ( ϕ 1 , Φ n ) L 2 (Ω) Φ n , then, from Lemma 2.23 , we get ∥ e ϕ k (0) ∥ 2 H 1 Γ (Ω; w ) = k X n =1 ( ϕ 1 , Φ n ) 2 L 2 (Ω) λ n ≤ ∞ X n =1 ( ϕ 1 , Φ n ) 2 L 2 (Ω) λ n = ∥ ϕ 1 ∥ H 1 Γ (Ω; w ) ; 30 and from ( 2.16 ) and ( 2.17 ) and Lemma 2.23 we ha ve ∂ t e ϕ k (0) = ∂ tt ϕ k (0) = k X n =1 d 2 ϕ k n d t 2 (0)Φ n = k X n =1 f k n (0)Φ n − k X n =1 ϕ k n (0) λ n Φ n and ∥ ∂ t e ϕ k (0) ∥ 2 L 2 (Ω) ≤ 2 ∥ f k (0) ∥ 2 L 2 (Ω) + 2 ∥ ϕ 0 ∥ 2 D ( A ) ≤ C ∥ f ∥ 2 H 1 (0 ,T ; L 2 (Ω)) + 2 ∥ ϕ 0 ∥ 2 D ( A ) , where the second inequalit y we use [ 8 , Theorem 2 (iii) in Chapter 5.9.2, p. 302], and the constan t C > 0 dep ends only on T . Hence, ess sup t ∈ [0 ,T ]  ∥ ∂ t e ϕ k ∥ 2 L 2 (Ω) + ∥ e ϕ k ∥ 2 H 1 Γ (Ω; w )  ≤ C  ∥ f ∥ 2 H 1 (0 ,T ; L 2 (Ω)) + ∥ ϕ 0 ∥ 2 D ( A ) + ∥ ϕ 1 ∥ 2 H 1 Γ (Ω; w )  (2.29) b y Lemma 2.23 , where the constant C > 0 dep ends only on T . Multiplying λ n ϕ k n ( t ) on the b oth sides of ( 2.19 ), summing n = 1 , · · · , k , note that ϕ k , A ϕ k = P k n =1 ϕ k n ( t ) λ n Φ n ∈ H 2 (Ω; w ) ∩ H 1 Γ (Ω; w ) for a.e. t ∈ [0 , T ] b y ( 2.16 ), from Theorem 2.17 (ii), we get ∥A ϕ k ∥ 2 L 2 (Ω) = ( ϕ k , A ϕ k ) H 1 Γ (Ω; w ) = ( f k − ∂ tt ϕ k , A ϕ k ) L 2 (Ω) , then, from ( 2.29 ) and [ 8 , Theorem 2 (iii) in Chapter 5.9.2, p. 302], we deduce ∥A ϕ k ( t ) ∥ L 2 (Ω) ≤ 2 ∥ f k ( t ) ∥ L 2 (Ω) + 2 ∥ ∂ tt ϕ k ( t ) ∥ L 2 (Ω) ≤ C  ∥ f ∥ H 1 (0 ,T ; L 2 (Ω)) + ∥ ϕ 0 ∥ D ( A ) + ∥ ϕ 1 ∥ H 1 Γ (Ω; w )  , (2.30) where the constant C > 0 dep ends only on T . Cho osing ζ ∈ C ∞ 0 ( R N ) , 0 < ζ < 1 , (2.31) suc h that ζ = 0 on T ×  0 , 1 4  , ζ = 1 on T ×  1 2 , 1  , and |∇ ζ | ≤ C and     ∂ 2 ζ ∂ x i ∂ x j     ≤ C on R N , where the constan ts C > 0 are absolute. Note that z = ζ ϕ k is the solution of the follo wing equation ( A z = ζ f k − ζ ∂ tt ϕ k − ϕ k A ζ − 2 ∇ ζ · A ∇ ϕ k , in T ×  1 4 , 1  , z = 0 , on T ×  1 4 , 1  , 31 this is a uniformly elliptic equation on T × ( 1 4 , 1), hence, from [ 8 , Theorem 1 (p. 327) in Chapter 6.3.1 and Theorem 4 (p. 334) in Chapter 6.3.2] and ( 2.14 ) and ( 2.20 ), we hav e ∥ ϕ k ( t ) ∥ H 2 (Ω 1 2 ) ≤ C  ∥ f k ( t ) ∥ L 2 (Ω) + ∥ ∂ tt ϕ k ( t ) ∥ L 2 (Ω) + ∥ ϕ k ( t ) ∥ H 1 Γ (Ω; w )  ≤ C  ∥ f ∥ H 1 (0 ,T ; L 2 (Ω)) + ∥ ϕ 0 ∥ D ( A ) + ∥ ϕ 1 ∥ H 1 Γ (Ω; w )  , (2.32) where the constan ts C > 0 dep end only on α and T . This, combing ( 2.22 ) and ϕ = ψ , we ha ve prov ed the 1th-4th estimate in ( 2.15 ). Step 7 . Hidden regularity . W e will pro v e the 4th estimate in ( 2.14 ) and the 5th estimate in ( 2.15 ). Let ϕ 0 ∈ D ( A ) , ϕ 1 ∈ H 1 Γ (Ω; w ) and f ∈ H 1 (0 , T ; L 2 (Ω)). Since ϕ is the weak solution of ( 1.1 ) with resp ect to ( ϕ 0 , ϕ 1 , f ), we obtain Ψ = ∂ t ϕ ∈ L 2 (0 , T ; H 1 Γ (Ω; w )) is the w eak solution of the following system            ∂ tt Ψ + A Ψ = ∂ t f , in Q, Ψ = 0 , on Γ × (0 , T ) , ∂ Ψ ∂ ν A = 0 , on Γ ∗ × (0 , T ) , Ψ(0) = ϕ 1 , ∂ t Ψ(0) = f (0) − A ϕ 0 , in Ω b y ( 2.15 ). Supp ose that the 4th estimate in ( 2.14 ) holds, then the 5th estimate in ( 2.15 ) also follows. So, we only need to prov e the 4th estimate in ( 2.14 ). T o prov e the 4th estimate in ( 2.14 ), by the density argument, w e only need to pro ve the case: y 0 ∈ D ( A ) , y 1 ∈ H 1 Γ (Ω; w ) and f ∈ H 1 (0 , T ; L 2 (Ω)). Let ζ b e defined in ( 2.31 ). Multiplying ζ n · ∇ ϕ on the b oth sides of ( 1.1 ), from ( 2.15 ), in tegrating on Q (indeed, on ( T × ( 1 4 , 1)) × (0 , T )), integration b y parts, we hav e A 1 + A 2 = Z Z Q ( ∂ tt ϕ )( ζ n · ∇ ϕ )d z d t − Z Z Q [div( A ∇ ϕ )]( ζ n · ∇ ϕ )d z d t = Z Z Q f ( ζ n · ∇ ϕ )d z d t = A 3 . F rom ∂ t ϕ = 0 on Γ × (0 , T ) and ζ = 0 on T × [0 , 1 4 ] and the 1th-2th estimate in ( 2.14 ), we ha ve A 1 = Z Ω ( ∂ t ϕ )( ζ n · ∇ ϕ )d z     t = T t =0 + 1 2 Z Z Q ( ∂ t ϕ ) 2 div( ζ n )d z d t ≤ C ess sup t ∈ [0 ,T ]  ∥ ∂ t ϕ ( t ) ∥ 2 L 2 (Ω) + ∥ ϕ ( t ) ∥ 2 H 1 Γ (Ω; w )  + C ∥ ∂ t ϕ ∥ 2 L 2 ( Q ) ≤ C  ∥ ϕ 0 ∥ 2 H 1 Γ (Ω; w ) + ∥ ϕ 1 ∥ 2 L 2 (Ω) + ∥ f ∥ 2 L 2 ( Q )  ; 32 from Lemma 2.25 we hav e (with n = ( n θ , n r )) A 2 = − 1 2 Z Z Γ × (0 ,T )  ∂ ϕ ∂ ν  2 d S d t + Z Z Q ( A ∇ ϕ ) · [( D ( ζ n )) ∇ ϕ ]d z d t − 1 2 Z Z Q ( ∇ ϕ · A ∇ ϕ ) div( ζ n )d z d t − α 2 Z Z Q ζ n r r α − 1 ( ∂ r u ) 2 d z d t ; from the 1th-2th estimate in ( 2.14 ), we ha v e A 3 ≤ ∥ f ∥ 2 L 2 ( Q ) + C ∥ ϕ ∥ 2 L 2 (0 ,T ; H 1 Γ (Ω; w )) ≤ C  ∥ ϕ 0 ∥ 2 H 1 Γ (Ω; w ) + ∥ ϕ 1 ∥ 2 L 2 (Ω) + ∥ f ∥ 2 L 2 ( Q )  , hence, Z Z Γ × (0 ,T )  ∂ ϕ ∂ ν  2 d S d t ≤ C  ∥ ϕ 0 ∥ 2 H 1 Γ (Ω; w ) + ∥ ϕ 1 ∥ 2 L 2 (Ω) + ∥ f ∥ 2 L 2 ( Q )  , (2.33) where the constant C > 0 dep ending only on α and T . This pro ves the 4th estimate in ( 2.14 ). Step 8 . F rom Step 1-Step 6 and Step 7, we get ( 2.14 ). F rom Step 6 and Step 7, w e get ( 2.15 ). W e complete the pro of of this theorem. The next lemma concerns the regularity of w eak solutions of ( 1.1 ). Lemma 2.27. L et ϕ 0 ∈ D ( A ) and ϕ 1 ∈ H 1 Γ (Ω; w ) and f ∈ H 1 (0 , T ; L 2 (Ω)) , and ϕ b e the we ak solution of ( 1.1 ) with r esp e ct to ( ϕ 0 , ϕ 1 , f ) . Then r ( ∂ t ϕ ) 2 = 0 on Γ ∗ × (0 , T ) . (2.34) Pr o of. F rom the third term in ( 2.15 ) and the b oundary condition ∂ t ϕ = 0 on Γ × (0 , T ), we obtain ∂ t ϕ ∈ L 2 (0 , T ; H 1 Γ (Ω; w )) and      Z Z Γ ∗ δ 1 × (0 ,T ) r ( ∂ t ϕ ) 2 d S d t − Z Z Γ ∗ δ 2 × (0 ,T ) r ( ∂ t ϕ ) 2 d S d t      =      Z Z Q δ 1 − Q δ 2 ∂ r  r ( ∂ t ϕ ) 2  d S d t      =      Z Z Q δ 1 − Q δ 2 ( ∂ t ϕ ) 2 d z d t + 2 Z Z Q δ 1 − Q δ 2 r ( ∂ t ϕ )( ∂ t ∂ r ϕ )d z d t      ≤ C Z Z Q δ 1 − Q δ 2 ( ∂ t ϕ ) 2 d z d t + C Z Z Q δ 1 − Q δ 2 r α ( ∂ r ∂ t ϕ ) 2 d z d t for all 0 ≤ δ 1 ≤ δ 2 ≤ 1 8 . This shows that Z Z Γ ∗ δ × (0 ,T ) r ( ∂ t ϕ ) 2 d S d t is contin uous for δ ∈ [0 , 1 8 ]. No w, from ∂ t ϕ = 0 on Γ × (0 , T ) and Lemma 2.3 and the third 33 term in ( 2.15 ), we get Z Z Γ ∗ δ × (0 ,T ) r ( ∂ t ϕ ) 2 d S d t = δ 1 − α 2 Z Z Q δ ∂ r  r α 2 ( ∂ t ϕ ) 2  d z d t = α 2 δ 1 − α 2 Z Z Q δ r α 2 − 1 ( ∂ t ϕ ) 2 d z d t + δ 1 − α 2 Z Z Q δ r α 2 ( ∂ t ϕ )( ∂ r ∂ t ϕ )d z d t ≤ C δ 1 − α 2 Z Z Q δ r α ( ∂ r ∂ t ϕ ) 2 d z d t + C δ 1 − α 2 Z Z Q δ ( ∂ t ϕ ) 2 d z d t → 0 as δ → 0. This completes the pro of of this lemma. Corollary 2.28. L et ϕ 0 ∈ D ( A ) , ϕ 1 ∈ H 1 Γ (Ω; w ) and f ∈ H 1 (0 , T ; L 2 (Ω)) . L et ϕ b e the solution of ( 1.1 ) with r esp e ct to ( ϕ 0 , ϕ 1 , f ) . Then ∂ θθ ϕ ∈ L 2 ( Q ) , and r α 2 ∂ θr ϕ ∈ L 2 ( Q ) , and r 1+ α 2 ∂ rr ϕ ∈ L 2 ( Q ) . (2.35) Mor e over, ther e exists a p ositive c onstant C , dep ending only on α , such that Z Z Q  ( ∂ θθ ϕ ) 2 + r α ( ∂ θr ϕ ) 2 + r 2+ α ( ∂ rr ϕ ) 2  d z d t ≤ C  ∥ ϕ 0 ∥ 2 D ( A ) + ∥ ϕ 1 ∥ 2 H 1 Γ (Ω; w ) + ∥ f ∥ 2 H 1 (0 ,T ; L 2 (Ω))  . (2.36) Pr o of. Note that ϕ satisfies ∂ tt ϕ + A ϕ = f , i.e., A ϕ = f − ∂ tt ϕ ∈ L 2 (Ω) for a.e. t ∈ [0 , T ]. F rom Theorem 2.20 w e get ( 2.35 ), and from ( 2.9 ) we get Z Z Q  ( ∂ θθ ϕ ) 2 + r α ( ∂ θr ϕ ) 2 + r 2+ α ( ∂ rr ϕ ) 2  d z d t ≤ C  ∥ ∂ tt ϕ ∥ 2 L 2 ( Q ) + ∥ f ∥ 2 L 2 ( Q )  ≤ C  ∥ ϕ 0 ∥ 2 D ( A ) + ∥ ϕ 1 ∥ 2 H 1 Γ (Ω; w ) + ∥ f ∥ 2 H 1 (0 ,T ; L 2 (Ω))  , where we use ( 2.15 ) in the last inequalit y . W e complete the pro of of this corollary . Corollary 2.29. L et ϕ 0 ∈ D ( A ) , ϕ 1 ∈ H 1 Γ (Ω; w ) and f ∈ H 1 (0 , T ; L 2 (Ω)) . L et ϕ b e the we ak solution to e quation ( 1.1 ) with r esp e ct to ( ϕ 0 , ϕ 1 , f ) . Then we have (i) ∂ θ ϕ = 0 on Γ × (0 , T ) , (ii) for any  ∈ [0 , 2 − α 4 ) , we have r α 2 ϕ 2 ∈ L 2 (0 , T ; W 1 , 1 (Ω)) , and r α 2 + ϵ ϕ ∈ L 2 (0 , T ; H 1 (Ω)) , and rϕ 2 = r ϕ = 0 on Γ ∗ × (0 , T ) , (iii) r ( ∇ ϕ · A ∇ ϕ ) ∈ L 2 (0 , T ; W 1 , 1 (Ω)) , and r ( ∇ ϕ · A ∇ ϕ ) = 0 on Γ ∗ × (0 , T ) . Pr o of. This is a direct consequence of Lemma 2.21 . Corollary 2.30. L et ϕ 0 ∈ D ( A ) and ϕ 1 ∈ H 1 Γ (Ω; w ) and f = 0 . L et ϕ b e the we ak solution of ( 1.1 ) with r esp e ct to ( ϕ 0 , ϕ 1 , f = 0) . Then, for e ach t ∈ [0 , T ] , we have E ( t ) = E (0) , with E ( t ) = 1 2 Z Ω  ( ∂ t ϕ ) 2 + ∇ ϕ · A ∇ ϕ  d z . 34 Pr o of. Multiplying ( 1.1 ) b y ∂ t ϕ , from Theorem 2.26 and Theorem 2.17 (ii), in tegrating on Q , we get 0 = Z Z Q [ ∂ tt ϕ − div( A ∇ ϕ )] ∂ t ϕ d z d t = 1 2 Z Z Q ∂ t ( ∂ t ϕ ) 2 d z d t + 1 2 Z Z Q ∂ t ( ∇ ϕ · A ∇ ϕ )d z d t = 1 2 Z Ω  ( ∂ t ϕ ) 2 + ( ∇ ϕ · A ∇ ϕ )  d z     t = T t =0 . This prov es the corollary . 3 Observ abilit y Let δ 0 ∈ (0 , 1 32 ) b e a given constant. Set I ω = [0 , 4 δ 0 ) ∪ (2 π − 4 δ 0 , 2 π ) , ω = I ω × (0 , 1) . The interior observ ation region is therefore the strip ω , and T \ I ω = (4 δ 0 , 2 π − 4 δ 0 ) . In the argumen t below, T \ I ω is treated inside a single co ordinate c hart of the smo oth manifold T ; more precisely , this set is contained in the c hart ( δ 0 , 2 π − δ 0 ). 3.1 A motiv ating example W e no w explain why the cut-off decomp osition introduced b elo w is natural, and why the strip ω m ust app ear in the final estimate. Cho ose χ ∈ C ∞ 0 (0 , 2), χ ≡ 0, and for ε ∈ (0 , 1 4 ) define η ε ( r ) = χ ( r /ε ) . Then η ε is supp orted in (0 , 2 ε ), so that η ε (1) = 0 , η ′ ε (1) = 0 . F or eac h in teger n ≥ 1, consider the oscillatory profile u n,ε ( θ , r , t ) = η ε ( r ) sin n ( θ − t ) . This family is not intended as a literal counterexample to the homogeneous observ ability statemen t. Instead, it captures the quasimo de-type obstruction that motiv ates the present pro of. 35 One has ∂ tt u n,ε = − n 2 η ε ( r ) sin n ( θ − t ) , ∂ θθ u n,ε = − n 2 η ε ( r ) sin n ( θ − t ) , hence the time and angular contributions cancel exactly: ∂ tt u n,ε − ∂ θθ u n,ε = 0 . Therefore ∂ tt u n,ε − div( A ∇ u n,ε ) = − ∂ r ( r α η ′ ε ( r )) sin n ( θ − t ) . Since η ε is supp orted near r = 0, this residual is concentrated near the degenerate b oundary . On the other hand, the top-b oundary observ ation is completely blind to this family: u n,ε ( θ , 1 , t ) = 0 , ∂ r u n,ε ( θ , 1 , t ) = η ′ ε (1) sin n ( θ − t ) = 0 . Th us this family exhibits the basic obstruction mec hanism: strong angular oscillation with negligible radial p enetration can remain essentially in visible from the top b oundary while the defect is confined near the degenerate side. In this sense, u n,ε should b e read as a source- supp orted quasimo de mec hanism rather than as an exact homogeneous counterexample. There is a second obstruction. The multiplier used b elo w is H ( θ , r ) = (( θ − π ) , r ) , whic h is not globally defined on the p erio dic v ariable θ ∈ T . Hence the m ultiplier metho d cannot b e applied directly on the whole cylinder. T o ov ercome this top ological difficult y , w e remo ve a narrow angular strip I ω = [0 , 4 δ 0 ) ∪ (2 π − 4 δ 0 , 2 π ) , and choose a cut-off ζ = ζ ( θ ) suc h that ζ ≡ 1 a wa y from I ω and ζ ≡ 0 near I ω . W e then decomp ose the solution as ϕ = ψ + ξ , ψ = ζ ϕ, ξ = (1 − ζ ) ϕ. The roles of these tw o pieces are transparen t: • ψ is supp orted in a single chart of T , so the multiplier argument with H = (( θ − π ) , r ) b ecomes legitimate; • ξ is supp orted in the strip ω = I ω × (0 , 1), whic h is exactly where the lo calization error pro duced b y the cut-off is concentrated. Therefore the strip ω is not an artificial add-on. It is forced b y the geometry of the quasimo de-t yp e obstruction and by the necessity of lo calizing the m ultiplier argument on 36 the p erio dic v ariable. This is wh y the final observ abilit y inequality prov ed b elo w takes a mixed form, inv olving the b oundary term on Γ and the interior observ ation on ω . 3.2 Cut-off decomp osition T aking ζ = ζ ( θ ) ∈ C ∞ ( R ) , 0 ≤ ζ ≤ 1 such that ζ = 1 on (3 δ 0 , 2 π − 3 δ 0 ) , ζ = 0 on ( −∞ , 2 δ 0 ) ∪ (2 π − 2 δ 0 , + ∞ ) , (3.1) and | ζ ′ | < C δ − 1 0 , | ζ ′′ | < C δ − 2 0 , where the p ositiv e constants C are absolute. The lo calized comp onen t ψ = ζ ϕ solves the follo wing system:            ∂ tt ψ − div ( A ∇ ψ ) = − 2 ∇ ζ · A ∇ ϕ − ϕ div( A ∇ ζ ) ≡ g , in b Q, ψ = 0 , on ∂ b Q − b Γ ∗ × (0 , T ) , ∂ ψ ∂ ν A = 0 , on b Γ ∗ × (0 , T ) , ψ (0) = ζ ϕ 0 , ∂ t ψ (0) = ζ ϕ 1 , in b Ω , (3.2) where b Ω = ( δ 0 , 2 π − δ 0 ) × (0 , 1), b Q = b Ω × (0 , T ), and b Γ ∗ = ( δ 0 , 2 π − δ 0 ) × { 0 } . The remainder ξ = (1 − ζ ) ϕ solves            ∂ tt ξ − div ( A ∇ ξ ) = 2 ∇ ζ · A ∇ ϕ + ϕ div( A ∇ ζ ) = − g , in e Q, ξ = 0 , on ∂ e Q − e Γ ∗ × (0 , T ) , ∂ ξ ∂ ν A = 0 , on e Γ ∗ × (0 , T ) , ξ (0) = (1 − ζ ) ϕ 0 , ∂ t ξ (0) = (1 − ζ ) ϕ 1 , in e Ω , (3.3) where e Ω = ([0 , 4 δ 0 ) ∪ (2 π − 4 δ 0 , 2 π )) × (0 , 1), e Q = e Ω × (0 , T ), and e Γ ∗ = ([0 , 4 δ 0 ) ∪ (2 π − 4 δ 0 , 2 π )) × { 0 } . Th us e Ω = ω . Since f = 0 ∈ H 1 (0 , T ; L 2 (Ω)), Theorem 2.26 gives ϕ ∈ L 2 (0 , T ; D ( A )) ∩ H 1 (0 , T ; H 1 Γ (Ω; w )) ∩ H 2 (0 , T ; L 2 (Ω)) , and hence ψ ∈ L 2 (0 , T ; D ( A )) ∩ H 1 (0 , T ; H 1 Γ (Ω; w )) ∩ H 2 (0 , T ; L 2 (Ω)) . (3.4) Moreo ver, Corollary 2.28 yields ∂ θθ ψ , r α 2 ∂ θr ψ , r 1+ α 2 ∂ rr ψ ∈ L 2 ( Q ) , (3.5) 37 and then ∂ θ ψ ∈ H 1 Γ (Ω; w ) , and r α ∂ r ψ ∈ H 1 (Ω; w ) . (3.6) Recall ( 3.1 ), we get ψ = 0 on [( δ 0 , 2 δ 0 ) × (0 , 1) × (0 , T )] ∪ [(2 π − 2 δ 0 , 2 π − δ 0 ) × (0 , 1) × (0 , T )] . (3.7) 3.3 Preliminary iden tities In this subsection we record the elemen tary identities needed in the m ultiplier com- putation. They are consequences of the supp ort prop erties of ψ , the regularity stated in ( 3.4 )–( 3.6 ), and rep eated applications of F ubini’s theorem. Lemma 3.1. The fol lowing identities hold: Pr o of. Since ψ ∈ L 2 (0 , T ; H 1 Γ (Ω; w )), w e first obtain: (I) ψ 2 ∈ L 1 ( b Q ), and ∂ θ ψ 2 = 2 ψ ∂ θ ψ ∈ L 1 ( b Q ), whic h shows that ψ 2 ∈ C 0 ( δ 0 , 2 π − δ 0 ) for a.e. ( r, t ) ∈ (0 , 1) × (0 , T ) by ( 3.7 ), and Z Z b Q ∂ θ ψ 2 d z d t = Z T 0 Z 1 0 Z 2 π − δ 0 δ 0 ∂ θ ψ 2 d θ d r d t = 0 (3.8) b y F ubini’s Theorem and ( 3.7 ); (I I) ψ 2 ∈ L 1 ( b Q ), and ∂ r ( r ψ 2 ) = ψ 2 + 2 r ψ ∂ r ψ ∈ L 1 ( b Q ), whic h shows that r ψ 2 ∈ C 0 [0 , 1] for a.e. ( θ , t ) ∈ ( δ 0 , 2 π − δ 0 ) × (0 , T ) by Corollary 2.29 (ii), and Z Z b Q ∂ r ( r ψ 2 )d z d t = Z T 0 Z 2 π − δ 0 δ 0 Z 1 0 ∂ r ( r ψ 2 )d r d θ d t = 0 (3.9) b y F ubini’s Theorem and Corollary 2.29 . Next, since ∂ θ ψ ∈ H 1 Γ (Ω; w ), w e hav e: (I I I) ( ∂ θ ψ ) 2 ∈ L 1 ( b Q ), and ∂ θ ( ∂ θ ψ ) 2 = 2 ∂ θ ψ ∂ θθ ψ ∈ L 1 ( b Q ), which shows that ( ∂ θ ψ ) 2 ∈ C 0 ( δ 0 , 2 π − δ 0 ) for a.e. ( r, t ) ∈ (0 , 1) × (0 , T ) b y ( 3.7 ), and Z Z b Q ∂ θ ( ∂ θ ψ ) 2 d z d t = Z T 0 Z 1 0 Z 2 π − δ 0 δ 0 ∂ θ ( ∂ θ ψ ) 2 d θ d r d t = 0 (3.10) b y F ubini’s Theorem and ( 3.7 ); (IV) ( ∂ θ ψ ) 2 ∈ L 1 ( b Q ), and ∂ r [ r ( ∂ θ ψ ) 2 ] = ( ∂ θ ψ ) 2 + 2 r ( ∂ θ ψ )( ∂ rθ ψ ) ∈ L 1 ( b Q ) by ( 3.5 ), which sho ws that r ( ∂ θ ψ ) 2 ∈ C 0 [0 , 1] for a.e. ( θ , t ) ∈ ( δ 0 , 2 π − δ 0 ) × (0 , T ) by Corollary 2.29 (i) and (iii), and Z Z b Q ∂ r  r ( ∂ θ ψ ) 2  d z d t = Z T 0 Z 2 π − δ 0 δ 0 Z 1 0 ∂ r  r ( ∂ θ ψ ) 2  d θ d r d t = 0 (3.11) 38 b y F ubini’s Theorem and Corollary 2.29 (i) and (iii) and α ∈ [1 , 2). (V) Since r α ( ∂ r ψ ) 2 ∈ L 1 ( b Q ) and ∂ θ [ r α ( ∂ r ψ ) 2 ] = 2 r α ( ∂ r ψ )( ∂ θr ψ ) ∈ L 1 ( b Q ) by ( 3.5 ), whic h sho ws that r α ( ∂ r ψ ) 2 ∈ C 0 ( δ 0 , 2 π − δ 0 ) for a.e. ( r, t ) ∈ (0 , 1) × (0 , T ), and Z Z b Q ∂ θ  r α ( ∂ r ψ ) 2  d z d t = Z T 0 Z 1 0 Z 2 π − δ 0 δ 0 ∂ θ  r α ( ∂ r ψ ) 2  d θ d r d t = 0 (3.12) b y F ubini’s Theorem and ( 3.7 ). (VI) Since r α +1 ( ∂ r ψ ) 2 ∈ L 1 ( b Q ) and ∂ r [ r α +1 ( ∂ r ψ ) 2 ] = ( α +1) r α ( ∂ r ψ ) 2 +2 r α +1 ( ∂ r ψ )( ∂ rr ψ ) ∈ L 1 ( b Q ) by ( 3.5 ), whic h sho ws that r α +1 ( ∂ r ψ ) 2 ∈ C [0 , 1] for a.e. ( θ , t ) ∈ ( δ 0 , 2 π − δ 0 ) × (0 , T ), and Z Z b Q ∂ r  r α +1 ( ∂ r ψ ) 2  d z d t = Z T 0 Z 2 π − δ 0 δ 0 Z 1 0 ∂ r  r α +1 ( ∂ r ψ ) 2  d r d θ d t = Z Z Γ × (0 ,T ) ( ∂ r ψ ) 2 d S d t (3.13) b y F ubini’s Theorem and Corollary 2.29 (iii) and α ∈ [1 , 2). 3.4 Multiplier argumen t Let the multiplier z 0 = ( π , 0) , H ( z ) = z − z 0 = ( θ − π , r ) , H · ∇ ψ = ( θ − π ) ∂ θ ψ + r ∂ r ψ . (3.14) W e first v erify that the multiplier is admissible. Note that Z Z b Q ( H · ∇ ψ ) 2 d z d t ≤ 2 Z Z b Q  ( θ − π ) 2 ( ∂ θ ψ ) 2 + r 2 ( ∂ r ψ ) 2  d z d t ≤ 2 π 2 Z Z b Q  ( ∂ θ ψ ) 2 + r α ( ∂ r ψ ) 2  d z d t < + ∞ , and from ∇ ( H · ∇ ψ ) = ( ∂ θ ψ + ( θ − π ) ∂ θθ ψ + r ∂ θr ψ , ( θ − π ) ∂ θr ψ + ∂ r ψ + r ∂ rr ψ ) (3.15) 39 w e get Z Z b Q [ ∇ ( H · ∇ ψ ) · A ∇ ( H · ∇ ψ )] d z d t = Z Z b Q [ ∂ θ ψ + ( θ − π ) ∂ θθ ψ + r ∂ θr ψ ] 2 d z d t + Z Z b Q r α [( θ − π ) ∂ θr ψ + ∂ r ψ + r ∂ rr ψ ] 2 d z d t ≤ C  Z Z b Q  ( ∂ θ ψ ) 2 + r α ( ∂ r ψ ) 2 + ( ∂ θθ ψ ) 2 + r α ( ∂ θr ψ ) 2 + r α +2 ( ∂ rr ψ ) 2  d z d t  < + ∞ b y Corollary 2.28 , these show that H · ∇ ψ ∈ H 1 (Ω; w ). W e now m ultiply ( 3.2 ) by H · ∇ ψ and in tegrate ov er b Q . This yields B 1 + B 2 ≡ Z Z b Q ( ∂ tt ψ )( H · ∇ ψ )d z d t − Z Z b Q [div( A ∇ ψ )]( H · ∇ ψ )d z d t = Z Z b Q g ( H · ∇ ψ )d z d t ≡ B 3 . W e next compute B 1 . F rom ( 2.15 ), w e hav e ∂ t ψ ∈ L 2 (0 , T ; H 1 Γ (Ω; w )), and therefore B 1 = Z Z b Q ∂ t [( ∂ t ψ )( H · ∇ ψ )] d z d t − Z Z b Q ( ∂ t ψ ) H · ∇ ∂ t ψ d z d t = Z b Q ( ∂ t ψ )( H · ∇ ψ )d z     t = T t =0 − 1 2 Z Z b Q [ ∇ ( ∂ t ψ ) 2 ] · H d z d t = Z b Q ( ∂ t ψ )( H · ∇ ψ )d z     t = T t =0 + 1 2 Z Z b Q ( ∂ t ψ ) 2 div H d z d t = Z b Q ( ∂ t ψ )( H · ∇ ψ )d z     t = T t =0 + Z Z b Q ( ∂ t ψ ) 2 d z d t b y Lemma 2.27 (or similarly ( 3.8 ) and ( 3.9 )). W e then compute B 2 . Since H · ∇ ψ ∈ H 1 (Ω; w ), Corollary 2.18 applies. Com bined with ( 3.7 ) and the identit y ∂ θ ψ = 0 on Γ × (0 , T ) (see Corollary 2.29 (i)), we obtain B 2 = − Z Z Γ × (0 ,T ) ( A ∇ ψ · ν )( H · ∇ ψ )d S d t + Z Z b Q A ∇ ψ · ∇ ( H · ∇ ψ )d z d t = − Z Z Γ × (0 ,T ) ( ∂ r ψ ) 2 d S d t + Z Z b Q A ∇ ψ · ∇ ( H · ∇ ψ )d z d t. 40 Using ( 3.15 ) together with ( 3.10 )-( 3.13 ), w e further compute Z Z b Q A ∇ ψ · ∇ ( H · ∇ ψ )d z d t = Z Z b Q  ( ∂ θ ψ ) 2 + r α ( ∂ r ψ ) 2  d z d t + 1 2 Z Z b Q  ( θ − π ) ∂ θ ( ∂ θ ψ ) 2 + ( θ − π ) r α ∂ θ ( ∂ r ψ ) 2 + r ∂ r ( ∂ θ ψ ) 2 + r α +1 ∂ r ( ∂ r ψ ) 2  d z d t = 1 2 Z Z Γ × (0 ,T ) ( ∂ r ψ ) 2 d S d t − α 2 Z Z b Q r α ( ∂ r ψ ) 2 d z d t. Then we get B 2 = − 1 2 Z Z Γ × (0 ,T ) ( ∂ r ψ ) 2 d S d t − α 2 Z Z b Q r α ( ∂ r ψ ) 2 d z d t. Com bining the formulae for B 1 and B 2 , we arrive at B 1 + B 2 = Z b Ω ( ∂ t ψ )( H · ∇ ψ )d z     t = T t =0 + Z Z b Q ( ∂ t ψ ) 2 d z d t − 1 2 Z Z Γ × (0 ,T ) ( ∂ r ψ ) 2 d S d t − α 2 Z Z b Q r α ( ∂ r ψ ) 2 d z d t. (3.16) W e now return to the equation for ψ . Multiplying ( 3.2 ) b y ∂ t ψ , and using ∂ t ψ ∈ L 2 (0 , T ; H 1 Γ (Ω; w )) together with Theorem 2.17 (ii), we get 1 2 Z b Ω  ( ∂ t ψ ) 2 + A ∇ ψ · ∇ ψ  d z     t = T t =0 = Z Z b Q g ( ∂ t ψ )d z d t. i.e., E ψ ( T ) − E ψ (0) = Z Z b Q g ( ∂ t ψ )d z d t, (3.17) where E ψ ( s ) = 1 2 Z b Ω  ( ∂ t ψ ) 2 + A ∇ ψ · ∇ ψ  d z     t = s . This implies that Z T 0 E ψ ( t )d t − T E ψ (0) = Z T 0 Z Z b Ω × (0 ,t ) g ( ∂ t ψ )d z d t, 41 i.e., T E ψ (0) = Z T 0 E ψ ( t )d t − Z T 0 Z Z b Ω × (0 ,t ) g ( ∂ t ψ )d z d t = 1 2 Z Z b Q  ( ∂ t ψ ) 2 + A ∇ ψ · ∇ ψ  d z d t − Z T 0 Z Z b Ω × (0 ,t ) g ( ∂ t ψ )d z d t. Then Z Z b Q ( ∂ t ψ ) 2 d z d t − α 2 Z Z b Q r α ( ∂ r ψ ) 2 d z d t = 2 − α 2 T E ψ (0) + 2 − α 2 Z T 0 Z Z b Ω × (0 ,t ) g ( ∂ t ψ )d z d t + 2 + α 4 Z Z Q ( ∂ t ψ ) 2 d z d t − 2 − α 4 Z Z Q ∇ ψ · A ∇ ψ d z d t − α 2 Z Z Q r α ( ∂ r ψ ) 2 d z d t. (3.18) On the other hand, m ultiplying ( 3.2 ) b y ψ and integrating o v er b Q , and using ψ ∈ L 2 (0 , T ; D ( A )) together with Theorem 2.17 (ii), we get Z Ω ( ∂ t ψ ) ψ d z     t = T t =0 = Z Z Q  ( ∂ t ψ ) 2 − ∇ ψ · A ∇ ψ  d z d t + Z Z b Q g ψ d z d t, and hence 1 2 Z Z Γ × (0 ,T ) ( ∂ r ψ ) 2 d S d t + Z Z b Q g ( H · ∇ ψ )d z d t − 2 − α 2 Z T 0 Z Z b Ω × (0 ,t ) g ( ∂ t ψ )d z d t + 2 + α 4 Z Z b Q g ψ d z d t ≥ 2 − α 2 T E ψ (0) + Z Ω ( ∂ t ψ )  H · ∇ ψ + 2 + α 4 ψ  d z d t     t = T t =0 (3.19) b y ( 3.16 ) and ( 3.18 ). Denote X ( t ) = Z Ω ( ∂ t ψ )  H · ∇ ψ + 2 + α 4 ψ  d z , for t ∈ [0 , T ] , hence, | X ( t ) | ≤  2 Z Ω ( ∂ t ψ ) 2 d z + 1 2  Z Ω     H · ∇ ψ + 2 + α 4 ψ     2 d z , for all  > 0 . Moreo ver, from Corollary 2.29 (ii), and ( 3.8 ) and ( 3.9 ), we get Z b Ω ( H · ∇ ψ ) ψ d z = 1 2 Z b Ω H · ∇ ψ 2 d z = − Z b Ω ψ 2 d z , 42 and therefore Z b Ω     H · ∇ ψ + 2 + α 4 ψ     2 d z = Z b Ω ( H · ∇ ψ ) 2 d z + 2 + α 2 Z b Ω ( H · ∇ ψ ) ψ d z + (2 + α ) 2 16 Z b Ω ψ 2 d z ≤ 2 Z b Ω ∇ ψ · A ∇ ψ d z b y α ∈ [1 , 2). Consequently , using B 1 + B 2 = B 3 , we get 1 2 Z Z Γ × (0 ,T ) ( ∂ r ψ ) 2 d S d t + Z Z b Q g ( H · ∇ ψ )d z d t − 2 − α 2 Z T 0 Z Z b Ω × (0 ,t ) g ( ∂ t ψ )d z d t + 2 + α 4 Z Z b Q g ψ d z d t ≥ 2 − α 2 T E ψ (0) −  2 Z Ω ( ∂ t ψ ) 2 d z − 1  Z Ω ∇ ψ · A ∇ ψ d z ≥  2 − α 2 T − max   2 , 1   E ψ (0) . Finally , from the definition of ψ (namely ψ = ζ ϕ ), from g = − 2 ∇ ζ · A ∇ ϕ − ϕ div( A ∇ ζ ) = − 2 ζ ′ ( θ ) ∂ θ ϕ − ζ ′′ ( θ ) ϕ, and from supp g ⊆  [2 δ 0 , 3 δ 0 ) ∪ (2 π − 3 δ 0 , 2 π − 2 δ 0 )  × (0 , 1) × (0 , T ) ⊆ ω × (0 , T ) , w e obtain | g | ≤ C ( | ϕ | + | ∂ θ ϕ | ) χ ω . Moreo ver, since ψ = ζ ϕ and H = (( θ − π ) , r ), w e hav e | ψ | + | ∂ t ψ | + | H · ∇ ψ | ≤ C ( | ϕ | + | ∂ t ϕ | + | ∂ θ ϕ | + r | ∂ r ϕ | ) on b Q . As α ∈ [1 , 2) and 0 < r < 1, one has r 2 ≤ r α . Hence, by Cauc hy–Sc hw arz and Y oung’s inequalit y ,     Z Z b Q g ( H · ∇ ψ )d z d t     ≤ C Z Z ω × (0 ,T ) ( | ϕ | + | ∂ θ ϕ | ) ( | ϕ | + | ∂ θ ϕ | + r | ∂ r ϕ | ) d z d t ≤ C Z Z ω × (0 ,T )  ϕ 2 + A ∇ ϕ · ∇ ϕ  d z d t, and similarly ,     Z T 0 Z Z b Ω × (0 ,t ) g ( ∂ t ψ )d z d t     +     Z Z b Q g ψ d z d t     ≤ C Z Z ω × (0 ,T )  ϕ 2 + ( ∂ t ϕ ) 2 + A ∇ ϕ · ∇ ϕ  d z d t. 43 Therefore, combining these b ounds with the previous inequalit y , we get h (2 − α ) T − √ 2 i E ψ (0) ≤ Z Z Γ × (0 ,T ) ( ∂ r ψ ) 2 d S d t + C Z Z ω × (0 ,T )  ϕ 2 + ( ∂ t ϕ ) 2 + A ∇ ϕ · ∇ ϕ  d z d t, (3.20) where the p ositiv e constant C dep ends only on α and δ 0 . It remains to estimate the lo calized comp onent ξ . W e multiply ( 3.3 ) by ∂ t ξ , where ∂ t ξ ∈ L 2 (0 , T ; H 1 Γ ( e Ω; w )) , and integrate by parts exactly as in ( 3.17 ). This yields E ξ (0) = E ξ ( t ) + Z Z e Ω × (0 ,t ) g ( ∂ t ξ )d z d t, where E ξ ( t ) = 1 2 Z e Ω  ( ∂ t ξ ) 2 + A ∇ ξ · ∇ ξ  d z d t, for t ∈ [0 , T ] . Since e Q = ω × (0 , T ), we get E ξ (0) ≤ 1 T Z T 0 E ξ ( t )d t +     Z Z e Q g ( ∂ t ξ )d z d t     ≤ C T Z Z ω × (0 ,T )  ϕ 2 + ( ∂ t ϕ ) 2 + A ∇ ϕ · ∇ ϕ  d z d t +     Z Z e Q g ( ∂ t ξ )d z d t     . Using again the b ound | g | ≤ C ( | ϕ | + | ∂ θ ϕ | ) χ ω and the identit y ∂ t ξ = (1 − ζ ) ∂ t ϕ , we obtain     Z Z e Q g ( ∂ t ξ )d z d t     ≤ C Z Z ω × (0 ,T ) ( | ϕ | + | ∂ θ ϕ | ) | ∂ t ϕ | d z d t ≤ C Z Z ω × (0 ,T )  ϕ 2 + ( ∂ t ϕ ) 2 + A ∇ ϕ · ∇ ϕ  d z d t. Therefore E ξ (0) ≤ C  1 + 1 T  Z Z ω × (0 ,T )  ϕ 2 + ( ∂ t ϕ ) 2 + A ∇ ϕ · ∇ ϕ  d z d t, where the p ositiv e constant C dep ends only on α and δ 0 . 44 W e now com bine the estimates for ψ and ξ . Since ϕ = ψ + ξ , we ha v e E (0) = Z Ω  ( ∂ t ϕ ) 2 + A ∇ ϕ · ∇ ϕ  d z     t =0 = Z Ω  ( ∂ t ψ + ∂ t ξ ) 2 + A ( ∇ ψ + ∇ ξ ) · ( ∇ ψ + ∇ ξ )  d z     t =0 ≤ 2 Z Ω  ( ∂ t ψ ) 2 + ( ∂ t ξ ) 2 + A ∇ ψ · ∇ ψ + A ∇ ξ · ∇ ξ  d z     t =0 = 2 E ψ (0) + 2 E ξ (0) , Com bining this inequality with ( 3.20 ), w e obtain h (2 − α ) T − √ 2 i E (0) ≤ C Z Z Γ × (0 ,T ) ( ∂ r ϕ ) 2 d S d t + C Z Z ω × (0 ,T )  ϕ 2 + ( ∂ t ϕ ) 2 + A ∇ ϕ · ∇ ϕ  d z d t, where the p ositiv e constant C dep ends on T , α , and δ 0 . 45 Ac kno wledgmen ts The authors w ould lik e to thank Sh ugeng Chai (Shanxi Univ ersity) and Y an He (Hub ei Univ ersity) for helpful discussions related to the observ ability problem considered in this pap er. References [1] J. Bai and S. Chai, Exact con trollability of wa v e equations with interior degener- acy and one- sided b oundary con trol, J. Syst. Sci. Complex. , 36(2023), 656-671. doi: 10.1007/s11424-023-1094-3 . [2] P . Cannarsa, P . Martinez, and J. 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