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APPEARED IN BULLETIN OF THE

arXiv:math/9201268v1 [math.AP] 1 Jan 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 1, Jan 1992, Pages 53-86SEMILINEAR WAVE EQUATIONSMichael StruweAbstract. We survey existence and regularity results for semi-linear wave equa-tions.

In particular, we review the recent regularity results for the u5-Klein Gordonequation by Grillakis and this author and give a self-contained, slightly simplifiedproof.Contents1. Introduction2.

Preliminaries3. Rauch’s result4.

Large data5. A remark on the super-critical case1.

IntroductionIn this survey we shall be interested in initial value problems for nonlinear waveequations of the type(1.1)utt −∆u + g(u) = 0in R3 × [0, ∞[,(1.2)ut=0 = u0,utt=0 = u1,where g : R →R and the initial data are given sufficiently smooth functions,and ut =∂∂tu, etc. The linear case g(u) = mu, where m ∈R, corresponds tothe classical Klein Gordon equation in relativistic particle physics; the constant mmay be interpreted as a mass and hence is generally assumed to be nonnegative.In an attempt to model also nonlinear phenomena like quantization, in the 1950sequations of type (1.1) with nonlinearities likeg(u) = mu + u3,m ≥0,were proposed as models in relativistic quantum mechanics with local interaction;see for instance Schiff[13] and Segal [14].Solutions could be real or complex-valued functions.

In the latter case it was natural to assume that the nonlinearitycommutes with the phase, that is, there holdsg(eiϕu) = eiϕg(u)for all ϕ ∈R,Received by the editors July 11, 1990 and, in revised form, December 7, 19901980 Mathematics Subject Classification (1985 Revision). Primary 35L05, 35A05, 35-02c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2MICHAEL STRUWEand hence, in particular, that g(0) = 0. In this case, g may be expressedg(u) = u f|u|2,which gives the form of equation (1.1) studied, for instance, by J¨orgens [8].

Here, forsimplicity, and since all important features of our problem already seem to exist inthis case, we confine ourselves to the study of real-valued solutions of equation (1.1).To model effects thought to arise in the case, for instance, of spinor fields u, thescalar equation (1.1) also has been considered in space dimensions n ≥3; see [14].Various other models involving nonlinearities g depending also on ut and ∇u,the spatial gradient of u, have been studied. The so-called “σ-model” involves anequation of type (1.1) for vector-valued functions subject to a certain (nonlinear)constraint.1 In this caseg(u) = u|ut|2 −|∇u|2,and the solution u = (u1, .

. .

, un) is constrained to satisfy the condition|u|2 = u21 + · · · + u2n = 1;see Shatah [15] for some recent results on this problem and references.To limit this survey to a reasonable length, however, we restrict our study to non-linearities depending only on u; that is, the semi-linear case. The examples statedpreviously suggest that we assume that g(0) = 0 and that g satisfies polynomialgrowth(1.3)g(u) ≤C(1 + |u|p−2)|u|for some p ≥2, C ∈R.Moreover, following Strauss [16, Theorem 3.1], we will assume that g satisfies theconditions(1.4)G(u) ≥−C|u|2for some C ∈R ,and(1.5)G(u)/g(u) →∞as |u| →∞,where G(u) =R u0 g(v)dv.

Let us briefly motivate the latter two conditions.First, (1.4) and (1.5) include the linear case (with no sign condition) or, moregenerally, the case of Lipschitz nonlinearities. Second, in the super-linear case, thatis, ifg(u)/|u| →∞as |u| →∞, conditions (1.4), (1.5) should be regarded asa coerciveness condition.

In fact, in this case finite propagation speed ≤1 andconservation of energy imply locally uniform a priori bounds in L2 for solutions of(1.1) in terms of the initial data; this will be developed in detail in §2.By contrast, in the noncoercive case it is easy to construct solutions of (1.1)with smooth initial data that blow up in finite time; for instance, for any α > 0 thefunctionu(x, t) =1(1 −t)α1In fact, as observed by Shatah and Tahvildar-Zadeh [21], under suitable symmetry assump-tions also σ-models give rise to semilinear wave equations of type (1.1) on R4 × R.

SEMILINEAR WAVE EQUATIONS3solves the equationutt −∆u = α(1 + α)u|u|2αand blows up at t = 1. Observe that for α =1m, m ∈N, the right member ofthis equation is analytic.

Modifying the initial data offx; |x| ≤2, say, we evenobtain a singular solution with C∞-data having compact support. (See John [7]for a blow-up result for a similar equation.) Thus, conditions like (1.3)–(1.5) seemnatural if we are interested in global solutions.The class (1.3)–(1.5) includes the following special cases(1.6)g(u) = mu|u|q−2 + u|u|p−2,m ≥0, 2 ≤q < p .As we shall see, for nonlinearities of this kind the answer to the existence problemfor (1.1), (1.2) in a striking way depends on the space dimension n and on theexponent p. In particular, in the physically interesting case n = 3, global existencefor p < 6 can be established with relative ease, while the same question for p > 6so far has eluded all research attempts.

The “critical” case p = 6 has only recentlybeen settled and a comprehensive account of this result is one of the objectivespursued in this survey.In fact, the apparent existence of a “critical power” for (1.1) and recent advanceson elliptic problems involving critical nonlinearities prompted our interest in theu5-Klein Gordon equation. “Critical powers” very often come into play in nonlinearproblems through Sobolev embedding.

In particular, p = 6 is the critical powerfor the Sobolev embedding H1,2loc (R3) ֒→Lploc(R3). (In n dimensions the criticalpower for this embedding is p =2nn−2.) Moreover, they very often arise naturallyfrom the requirements of scale invariance, that is, whenever “intrinsic” notions areinvolved.

A beautiful example of such a problem is the Yamabe problem concern-ing the existence of conformal metrics with constant scalar curvature on a given(compact) Riemannian manifold. Through the work of Trudinger, Aubin, and—finally—Schoen this problem has now been completely solved and it has becomeapparent that at the critical power properties like “compactness of the solution set”depend crucially on global aspects of the problem; in this case, on the topologicaland differentiable structure of the manifold.

See Lee and Parker [9] for a recentsurvey of the Yamabe problem in this journal.Incidentally, for nonlinear wave equations (or nonlinear Schr¨odinger equationsiut −∆u + u|u|p−2 = 0) there appear to be many “critical powers,” depending onwhat aspect of the problem we consider: global existence, scattering theory, . .

. ;see Strauss [16, p.

14f.]. As regards global existence, it remains to be seen whetherthe critical power represents only a technical barrier or, in fact, defines the dividingline between qualitatively different regimes of behavior of (1.1), (1.2).

Through thissurvey I would like to invite further research on this topic.We conclude this introduction with a short overview of the existence results inthe case of a pure power(1.7)utt −∆u + u|u|p−2 = 0,p > 2.For more general nonlinearities of type (1.6) similar results hold true. (In con-trast, for problems related to scattering, also the lower order terms of g may bedecisive.)

4MICHAEL STRUWEThe sub-critical case. For n = 3, p < 6 global existence and regularity wasestablished by J¨orgens [8] in 1961.

J¨orgens also was able to show local (small time)existence of regular solutions to (1.7), (1.2) for arbitrarily large p. Moreover, hewas able to reduce the problem of existence of global, regular solutions to (1.1) to(local) estimates of the L∞-norms of solutions.These results have been generalized to higher dimensions; however, such exten-sions have been very hard to obtain. While J¨orgens’ work relies on the classicalrepresentation formula for the 3-dimensional wave equation, this method fails inhigher dimensions n > 3.

The fundamental solution to the wave equation no longeris positive; moreover, it carries derivatives transverse to the wave cone. Neverthe-less, at least for n ≤9, the existence results of Pecher [11], Brenner-von Wahl [2]now cover the full sub-critical range p <2nn−2.

Regular solutions are unique.Global weak solutions. On the other hand, by a suitable approximation andusing energy estimates, for all p > 2, n ≥3 it is possible to construct global weaksolutions, satisfying (1.7) in a distributional sense; see Segal [14], Lions [10].

Inthis case, it even suffices to assume that the initial data u0, u1 ∈L2loc(Rn) withu0 ∈Lploc(Rn) and distributional derivative ∇u0 ∈L2loc(Rn).Energy estimatesimmediately give uniqueness of weak solutions in case p ≤2nn−2 −2n−2; see Brow-der [3]. However, this range is well below the critical Sobolev exponent p =2nn−2.

Inorder to improve the range of admissable exponents, more sophisticated tools weredeveloped, based, in particular, on the Lp −Lq-estimates for the wave operator byStrichartz [17]; see also Brenner [1]. In their simplest version, these estimates allowto prove uniqueness of solutions to (1.7), (1.2) for p ≤2(n+1)n−1 , the Sobolev exponentin (n + 1) space dimensions.

In fact, uniqueness can be established for p <2nn−2;see Ginibre-Velo [4]. In this case, moreover, the unique solution can be shown tobe “strong,” that is, to possess second derivatives in L2 and to satisfy the energyidentity [4].

Some of these results will be derived in §2.The critical case. In dimension n = 3, global existence of C2-solutions in thecritical case p = 6 was first obtained by Rauch [12], assuming the initial energyEu(0)=ZR3|u1|2 + |∇u0|22+ |u|66dxto be small.

His results will be presented in §3.In 1987, also for “large” data global C2-solutions were shown to exist by thisauthor [18] in the radially symmetric case u0(x) = u0|x|, u1(x) = u1|x|. Finally,Grillakis [6] in 1989 was able to remove the latter symmetry assumption, yieldingthe following result:Theorem 1.1.

For any u0 ∈C3(R3), u1 ∈C2(R3) there exists a unique solutionu ∈C2R3 × [0, ∞[to the Cauchy problem(1.8)utt −∆u + u5 = 0,(1.9)ut=0 = u0,utt=0 = u1.In §4 we present the detailed proof. Related partial regularity results independentlyhave been obtained by Kapitanskii [20] in 1989.Uniqueness holds among C2-solutions.

The proof procedes via a priori estimates. The classical representation

SEMILINEAR WAVE EQUATIONS5formula crucially enters. It seems unlikely that regularity or uniqueness of weaksolutions to (1.8), (1.9) can be established in a similar way.

Research on the criticalcase in higher dimensions is in progress; however, to this moment the results on thissubject still seem incomplete. Advances in these questions may require eliminatingthe use of the wave kernel.The super-critical case.

In §5 we observe that for sufficiently small initial datathe existence of global regular solutions, for instance, to the equationutt −∆u + u5 + u|u|p−2 = 0in R3 × [0, ∞[,for any p > 2 can be deduced as a corollary to Rauch’s result. Various qualita-tive properties of solutions in the super-critical case have recently been studied byZheng [19].Other open problems concern scattering theory, involving, in particular, decayestimates for solutions of (1.1) (see Ginibre-Velo [5]), or existence and regularityresults for initial-boundary value problems.2.

PreliminariesWe begin our study of (1.1) with some general comments about local solvabilityand global continuation of solutions to (1.1), (1.2). An excellent reference for manyfundamental results on nonlinear wave equations is Strauss [16]; our treatment ofthese issues will be somewhat narrower and directed towards our final goal: thecritical power.

This restricted aim, however, will enable us to make this paper es-sentially self-contained and to present a lot of material connected with the existenceproblem for (1.1), (1.2) in detail, introducing the reader to various approaches tothis problem and showing their strengths and limitations.Representation formulas. The representation of solutions to the inhomogeneouswave equation in terms of the fundamental solution and energy estimates form thebasis of our solution method.

For any f ∈C∞, u0, u1 ∈C∞there exists a uniqueC∞-solution to the Cauchy problem(2.1)utt −∆u = fin Rn × [0, ∞[,(2.2)ut=0 = u0,utt=0 = u1.If n = 3, the most interesting case, this solution, in fact, is given byu(x, t) = ddt 14πtZ∂Bt(x)u0(y) dy!+14πtZ∂Bt(x)u1(y) dy+ 14πZ t0Z∂Bt−s(x)f(y, s)t −s dy ds,(2.3)where Br(x) =y ∈Rn; |x−y| < r. From (2.3) we see immediately that informa-tion propagates with speed ≤1. In particular, u(t) has compact support for anyt ≥0 if this is the case for u0, u1, and f. However, (2.3) also shows a fundamentalweakness of the classical approach: For u0 ∈C3, u1 ∈C2, f ∈C2, the solution

6MICHAEL STRUWEu will lie in C2, only. That is, we encounter a loss of differentiability.

In higherdimensions, a representation formula similar to (2.3) holds, however, involving aneven larger number (the integer part of n2 , resp. n−22) of derivatives of u0, resp.

ofu1 and f. This makes the representation formula appear to be ill-suited for provingexistence of solutions for semilinear equations in dimensions n > 3.By contrast, no loss of differentiability will occur if instead of pointwise controlof the solution we are content with control of integral norms. The basic observationis the following.Energy inequality.

Upon multiplying (2.1) by ut we obtainddt|ut|2 + |∇u|22−div(ut∇u) = f ut,where the termse0(u) = |ut|2 + |∇u|22and p(u) = ut∇u may be interpreted as energy and momentum of the solution u.Integrating in x, if u(t) has compact support, by H¨older’s inequality we obtainddtE0u(t)≤ZRnf(·, t)2 dx1/2 ZRnut(·, t)2 dx1/2≤2E0u(t)1/2f(·, t)L2(Rn),whereE0u(t)=ZRn e0u(·, t)dx =:u(t)20denotes the “energy norm.”Thus(2.4)ddtu(t)0 ≤1√2f(·, t)L2(Rn) ≤f(·, t)L2(Rn).In particular, if f = 0, the “energy” E0 is conserved.Various other conservation laws can be obtained by using further multipliersrelated to symmetries of the wave operator. Very subtle identities and integralestimates in this way have been found; see Strauss [16, Chapter 2] for an overviewof results.

In particular, in §4 we will make use of the integral estimate implied byinvariance of the wave operator under dilations (x, t) 7→(R x, R t) for R > 0. Forour immediate uses, however, the energy inequality will suffice.So far, (2.4) has been established rigourously only for C∞-data u0, u1, and f withspatially compact support.

For our next topic it is essential to extend the validityof (2.4) to distribution solutions of (2.1) for finite energy initial data, that is, foru0, u1 ∈L2(Rn) with ∇u0 ∈L2(Rn), and functions f belonging to L2Rn × [0, T ]for any T > 0. To achieve this extension, by density of C∞0 (Rn) in L2(Rn) wemay approximate data u0, u1 as above by functions u(m)0, u(m)1∈C∞0 , convergingto u0, u1 in energy norm as m →∞.

Similarly, for any T > 0 we can find smooth,compactly supported functions f (m) converging to f in L2Rn × [0, T ]. Let u(m)

SEMILINEAR WAVE EQUATIONS7be the corresponding sequence of solutions to (2.1), (2.2), given by the classicalrepresentation formula.Then, applying (2.4) to the difference v = u(m) −u(l)of any two solutions, we see that u(m)(·, t) is a Cauchy sequence in energy norm,uniformly in t ∈[0, T ]. The limit u is a distribution solution to (2.1), (2.2) withuniformly finite energy in the interval [0, T ], which satisfies (2.4) in the slightlyweaker sense(2.5)u(t)0 ≤u(0)0 +Z t0f(·, s)L2(Rn) ds,for all t ≤T .

In particular, u is unique in this class.In a similar way, we now use (2.5) to construct solutions to nonlinear waveequations (1.1), (1.2) for smooth, compactly supported initial data and smoothnonlinearities satisfying a global Lipschitz condition by a contraction mapping ar-gument.Global solutions for Lipschitz nonlinearities. Indeed, if g : R →R is smoothand globally Lipschitz, for any v ∈C∞0Rn × [0, ∞[we obtain a C∞-solutionu = K(v) to the initial value problemutt −∆u = −g(v)with data u0, u1.

By (2.5), for all T > 0 we havesup0≤t≤TK(v) −K(˜v)(t)0 ≤Z T0g(v) −g(˜v)(t)L2 dt≤LZ T0(v −˜v)(t)L2 dt≤T L sup0≤t≤T(v −˜v)(t)L2,where L denotes the Lipschitz constant of g. Moreover, if u0, u1 have support inBR(0), and if v(t) has support in BR+t(0), so will u(t).Finally, by Poincar´e’sinequality, for such v, ˜v, and t ≤1 we can estimate(v −˜v)(t)L2 ≤(R + t)∇(v −˜v)(t)L2 ≤√2(R + 1)(v −˜v)(t)0.Thus, for T ≤minn1,1√2L(R+1)othe map K extends to a contracting map on thespaceV =nv ∈L2Rn × [0, T ]; suppv(t)⊂BR+t(0), vt(t), ∇v(t) ∈L2(Rn)for almost every t, andsup0≤t≤Tv(t)0 < ∞o,endowed with the norm||v||V = sup0≤t≤Tv(t)0.Let u be the unique fixed point of K in V ; then u weakly solves (1.1) and assumesits initial data (1.2) in the distribution sense. By an approximation argument as in

8MICHAEL STRUWEthe preceding paragraph, likewise for compactly supported measurable initial datawith finite energy we obtain a (unique) local solution to (1.1), (1.2) in the space V .Observe that the support of the solution grows with speed ≤1. Hence, given anynumber T0 > 0, by iterating the above construction a finite number of times (withT ≤minn1,1√2L(R+T0)o) we obtain a finite energy solution to (1.1), (1.2) on theinterval [0, T0] for any finite energy initial data supported in a ball of radius R.Since T0 is arbitrary, this solution can be continued globally.Finally, by finiteness of propagation speed also the assumption that the initialdata be compactly supported can be removed.

Indeed, if u and v solve (1.1) for com-pactly supported, finite energy initial data that coincide on the ball BR(0), their dif-ference ˜u will solve an equation of type (1.1) with a Lipschitz nonlinearity ˜g, where˜g(˜u) = g(u)−g(v), and initial data vanishing on BR(0). By the above existence anduniqueness result, ˜u is supported outside the light cone K =(x, t); |x| < R −tabove BR(0).

Hence a solution to (1.1) on K is entirely determined by its data inBR(0). For arbitrary data u0, u1 with locally finite energy, and k ∈N, we then letu(k)0 , u(k)1be compactly supported data that agree with u0, u1 on Bk(0).

For anyk ∈N the corresponding global solutions u(k), k ≥k0, then agree on the cone aboveBk0(0). Hence the sequence (u(k)) converges locally in energy norm to a globalsolution u of (1.1), (1.2).

In the same way, as far as global existence is concerned,in the following for convenience—and with no loss of generality—we may supposethat the initial data have compact support. Moreover, for our next topics we alsorequire the data u0, u1 to be smooth.Strong solutions.

Taking difference quotients u(h) =u(·)−u(·+he)hin any spacedirection e and passing to the limit h →0 we see that v = e · ∇u weakly solvesvtt −∆v + g′(u)v = 0and satisfiesv(T )0 −v(0)||0 ≤Z T0∥g′(u)v(t)∥L2 dt ≤LZ T0v(t)L2 dt≤C LZ T0v(t)0 dt,(2.6)for any T > 0. Thus ∇ut(t), ∇2u(t) ∈L2(Rn), uniformly in t ∈[0, T ], and fromequation (1.1) it now also follows that utt(t) ∈L2, uniformly in t ∈[0, T ], for anyT > 0.

This is the class of “strong solutions” to nonlinear wave equations. Forstrong solutions we can derive the strong form of the energy inequality (2.4).

Sinceg(u)ut = ddtG(u), upon multiplying (1.1) by ut we obtain the conservation lawddt|ut|2 + |∇u|22+ G(u)−div(ut∇u) = 0,where the termeu(t)= |ut|2 + |∇u|22+ G(u) = e0(u) + G(u)

SEMILINEAR WAVE EQUATIONS9now also contains the “potential energy density” G(u). LetEu(t)=ZRn eu(t)dx.Integrating over Rn, since u(t) has compact support, we thus obtain that(2.7)ddtEu(t)= 0,and energy is strictly conserved.Higher regularity.

By iterating the above procedure, we may want to deriveL2-bounds for higher and higher derivatives Dku, where D denotes any space-timederivative, k ∈N0. For instance, in case k = 3, any second order spatial derivativew = ∇2u satisfieswtt −∆w + g′(u)w + g′′(u)|∇u|2 = 0.However, while the term involving g′(u)w can be dealt with as before, the secondterm presents some difficulty and can only be controlled in terms of E0(w) if thedimension n ≤8.

In this case, assuming |g′′(u)| ≤C, by Sobolev’s inequality wecan estimateg′′(u)∇u(t)2L2 ≤C∇u(t)2L4 ≤C∇2u(t)2γL2∇3u(t)2−2γL2≤C E0(w)1−γwhereγ2nn−2 + (1−γ)2nn−4 ≤14. The energy inequality (2.4) formally yieldsddtw(t)0 ≤Cw(t)0 + Cw(t)2−2γ0.Note that γ →0 as n ր 8.

Hence the last exponent may be > 1, andw(t)0might blow up in finite time. Similar problems arise if we want to control higherderivatives of u by this simple trick.If n ≥9, we cannot start our iteration at k0 = 2.

However, if we choose k0 ∈Nsufficiently large, by using Sobolev’s embedding theorem as above we can derive apriori estimates forDku(t)0 for any k ≥k0 in a small time interval 0 ≤t ≤T (k).As in low dimensions, these estimates also may blow up in finite time.Nevertheless, we can use these estimates to show the local (small time) existenceof solutions to general semilinear equations (1.1) for smooth, compactly supportedinitial data.Local solutions for semi-linear equations. Indeed, given an arbitrary smoothmap g : R →R we may approximate g by maps g(k) satisfying a uniform Lipschitzcondition and coinciding with g for |u| ≤k.By the preceding discussion, given any smooth initial data of compact support,for each k ∈N we obtain a global strong solution u(k) of the approximate equationutt −∆u + g(k)(u) = 0,withDlu(k)(t)0 ≤C for any l ∈{0, .

. .

, l0} on some interval 0 ≤t ≤T = T (l0),where C depends on the Lipschitz constant of g(k), l0, T , and the size of the supportof the initial data.

10MICHAEL STRUWEIf n = 3, by the Sobolev embedding theoremDlu(k)(t)L∞≤CD2+lu(k)(t)L2 ≤CD1+lu(k)(t)0,for l = 0, 1, 2. In particular, for large k ∈N and sufficiently small T > 0, we obtainu(k)(x, t) ≤k in R3 ×[0, T ], and u(k) will be a solution to (1.1).

Similarly, if n > 3we can boundDlu(k)(t)L∞in terms ofDm+lu(k)(t)||0, where m > n2 −1, and we may conclude as before.Again remark that by finiteness of propagation speed the assumption that theinitial data be compactly supported can be removed; in this case, however, we canonly assert the existence of a solution to (1.1), (1.2) in a neighborhood of Rn × {0}.Global weak solutions. We now specialize our nonlinearity g to be of the form(1.3)–(1.5).

By assumption (1.4) there exist sequences r±k →±∞as k →∞suchthatr±k gr±k≥−Cr±k2 .We approximate g by Lipschitz functionsg(k)(u) =g(r−k ),if u < r−k ,g(u),if r−k ≤u ≤r+k ,g(r+k ),if u > r+k ,with primitive G(k)(u). Note that the approximating functions g(k) satisfy (1.4)with a uniform constant C. Now, for any k ∈N and smooth, compactly supporteddata we obtain a unique global strong solution u(k) to the approximate problem(1.1), (1.2) with D2u(k)(t) ∈L2(Rn) for all t.Conservation of energy (2.7) implies uniform bounds for u = u(k).

LetE(k)u(t)= E0u(t)+ZRn G(k)u(t)dx.By (1.4), (2.7), for any t ≥0 we have(2.8)E0u(t)−Cu(t)2L2 ≤E(k)u(t)= E(k)u(0)≤C < ∞,uniformly in k ∈N. In order to controlu(t)L2, for fixed x ∈Rn we estimateu(x, t) −u0(x)2 =Z t0ut(x, s) ds2≤tZ t0ut(x, s)2 ds.Integrating in x, by Minkowski’s inequality we obtainu(t)L2 ≤||u0||L2 +2tZ t0E0u(s)ds1/2.For t ≤T this and (2.8) gives the integral inequalityE0u(t)≤C + CTZ t0E0u(s)ds

SEMILINEAR WAVE EQUATIONS11for E0u(t).From Gronwall’s lemma we thus conclude that u(t) = u(k)(t) isuniformly bounded in energy norm on any interval [0, T ], uniformly in k ∈N.Hence, (u(k))k∈N is weakly relatively compact in the energy norm. Moreover,the support of u(k)(t) is bounded uniformly in k, for all t ≤T .

By the Rellich-Kondrakov theorem, therefore, we may assume that u(k) →u strongly in L2(Q) onany compact space-time region Q and pointwise almost everywhere. The limit uhas finite energyE0u(t)≤lim infk→∞E0u(k)(t),and(2.9)ZRn Gu(t)dx ≤lim infk→∞ZRn G(k)u(k)(t)dxfor almost every t > 0, by Fatou’s lemma.

Finally, for ϕ ∈C∞0Rn×]0, ∞[weobtainZ Z(utϕt −∇u ∇ϕ) dx dt = limk→∞Z Z(u(k)tϕt −∇u(k)∇ϕ) dx dt= limk→∞Z Zg(k)(u(k))ϕ dx dt =Z Zg(u)ϕ dx dt,whereR R. . .

denotes integration over Rn×[0, T ]. That is, u weakly solves equation(1.1).

(Vitali’s theorem, (1.5) and (2.9) were used to pass to the limit in thenonlinear term.) Similarly, approximating L2-data u0, u1 of finite energyZRn|u1|2 + |∇u0|22+ G(u0)dx < ∞by functions u(k)0 , u(k)1∈C∞0 (Rn), the existence of global weak solutions to (1.1)for arbitrary finite energy data may be derived.Regularity and uniqueness.

In the special case (1.7) with p ≤2nn−2 −2n−2 energyestimates may be used to obtain higher regularity and uniqueness. Indeed, let u(h)be a difference quotient in direction e ∈Rn as before.

Then, upon passing to thelimit h ց 0, for v = e · ∇u we obtainvtt −∆v = (1 −p)|u|p−2v,and thus, formally, by H¨older’s inequality and (2.4), thatddtv(t)∥0 ≤C|u|p−2v(t)∥L2 ≤Cu(t)p−2L2∗v(t)L2∗,where 2∗=2nn−2. Sobolev’s inequality now implies thatddtv(t)0 ≤Cu(t)p−20v(t)0and it follows that E0v(t)< ∞for all t; that is, u is a strong solution to (1.1).Similarly, higher regularity (for small time, if the dimension is large) may be ob-tained.

To see uniqueness, let u, ˜u be solutions to (1.7) with the same initial data(1.2). For v = u −˜u we obtain the inequalityddtv(t)0 ≤Cu(t)0 +˜u(t)0p−2v(t)0.Since v(0) = vt(0) = 0, uniqueness follows.By more sophisticated methods the above regularity and uniqueness results maybe extended to the full sub-critical range p <2nn−2; see Ginibre-Velo [4].

One suchmethod will be briefly explained next.

12MICHAEL STRUWELp −Lq-estimates. By a result of Strichartz [17], for any Lp-solution of the waveequation in Rn × [0, ∞[ with □u = utt −∆u ∈Lpp−1 and vanishing initial data wehave(2.10)u(t)Lp ≤CZ t0(t −s)1−2nδ□u(s)Lpp−1 ds,provided δ = 12 −1p ≤1n+1; see also Brenner [1].

We illustrate how this estimate maybe used to obtain global strong (or even classical) solutions to (1.7) in dimensionsn > 3. Since we will need g ∈C2, we suppose that p ≥3.

The above condition onδ then requires n ≤5, p ≤2(n+1)n−1 .It suffices to show existence of a solution on Rn ×[0, T ] for compactly supported,smooth data and for arbitrary T > 0. Let u(0) solve u(0)tt −∆u(0) = 0 with initialdata (1.2), and for k ∈N let u(k) be the solution to the approximate equationu(k)tt −∆u(k) + u(k) min|u(k)|p−2, kp−2= 0,together with (1.2).

Here, min{a, b} is a smooth function coinciding with the min-imum of a and b for |a −b| ≥1.Then u(k) = u(0) + v(k), where(2.11)□v(k) = v(k)tt −∆v(k) = −u(k) min|u(k)|p−2, kp−2,withv(k)(0) = v(k)t(0) = 0.By (2.6), D2u(k)(t) ∈L2(Rn) for all t. Moreover, by (2.9) we can uniformly boundE0u(k)(t)≤C0,ZRn min(|u(k)|pp, kpp +|u(k)|2 −k2kp−22)dx ≤C0.That is, □v(k)(t) is uniformly bounded in Lpp−1 (Rn) for all t. From (2.10) we nowobtain(2.12)u(k)(t)Lp ≤u(0)(t)Lp +v(k)(t)Lp≤C(t) + CZ t0(t −s)1−2nδ□v(k)(s)Lpp−1 dx ≤C(T )for all t < T , since δ = 12 −1p ≤1n+1 < 1n for the range of p and n considered.Differentiating (2.11), similarly we obtain□(Dv(k)) ≤(p −1)|Du(k)| min|u(k)|p−2, kp−2≤(p −1)|Du(k)| |u(k)|p−2.Hence □(Dv(k)) ∈Lpp−1 and by (2.12)□(Dv(k))(t)Lpp−1 ≤(p −1)Du(k)(t)Lpu(k)(t)p−2Lp≤C(T )Du(k)(t)Lp.

SEMILINEAR WAVE EQUATIONS13Thus, from (2.10) we inferDu(k)(t)Lp ≤C(T ) + C(T )Z t0(t −s)1−2nδDu(k)(s)Lp ds,and it follows thatDu(k)(t)Lp ≤C(T ) for all 0 ≤t ≤T .Finally, we have□(D2v(k)) ≤C|Du(k)|2|u(k)|p−3 + |D2u(k)||u(k)|p−2,whence□(D2v(k))(t)Lpp−1 ≤CDu(k)(t)2Lpu(k)(t)p−3Lp + CD2u(k)(t)Lpu(k)(t)p−2Lp≤C(T )1 +D2u(k)(t)Lp,and from (2.10) again it follows thatD2u(k)(t)Lp ≤C(T ), uniformly in k. Butby Sobolev’s inequality, for n ≤5, p ≥3, we may estimateu(k)(t)L∞≤CD2u(k)(t)Lp ≤C(T ).It follows that for sufficiently large k the function u = u(k) is a (strong) solutionto the original equation (1.7). If g ∈C4, we can proceed to bound the first andsecond derivatives of u and hence obtain a classical solution.Note that the range p ≤2n+2n−1 , where Stichartz’ estimate may be applied, slightlyexceeds the range p ≤2nn−2 −2n−2, where simple energy estimates suffice to showregularity and uniqueness.Classical solutions.

If n = 3, using (2.3) one can also devise a contraction map-ping argument in the space C2 to obtain local classical solutions to (1.1), (1.2) forinitial data u0 ∈C3, u1 ∈C2 with compact support.Indeed, via (2.3) the initial value problem (1.1), (1.2) can be converted into theintegral equationu(x, t) = v(x, t) −14πZ t0Z|x−y|=t−sgu(y, s)t −sdy ds,where v denotes the solution to the homogeneous wave equation with data (1.2). Ifg is smooth and globally Lipschitz this can easily be solved on R3×[0, T ] for suitablysmall T > 0 by a contraction mapping argument in the space C0R3 × [0, T ]withthe L∞-norm.

Differentiating (1.1) in any spatial direction, similarly we obtainDu(x, t) = Dv(x, t) −14πZ t0Z|x−y|=t−sg′(u)Dut −sdy dsand an analogous equation for the second spatial derivatives, from which we canas usual derive locally uniform bounds for all first and second derivatives of u onR3 × [0, T ]. To extend u beyond T we writeu(x, t) = v1(x, t) −14πZ tTZ|x−y|=t−sgu(y, s)t −sdy ds,

14MICHAEL STRUWEwhere now v1 denotes the solution of the homogeneous wave equation with datau(·, T ) and ut(·, T ) at time T . At first it may seem as if we had lost one derivativein this procedure.

However, following J¨orgens [8, p. 301], we can writev1(x, t) = v(x, t) −14πZ T0Z|x−y|=t−sgu(y, s)t −sdy ds,and v1 ∈C2, as desired. Thus, for smooth Lipschitz nonlinearities by iteration weobtain global C2-solutions.

Likewise, for smooth g we obtain local C2-solutions (forsmall time). However, if g′(u) is uniformly bounded (for instance, if u is uniformlybounded) on any interval [0, T ], then also this solution extends globally.

Finally,by finiteness of propagation speed, the assumption that the data have compactsupport can be dropped.Due to loss of differentiability in the nonlinear term, in dimensions n > 3 thisapproach—apparently—fails.After this preliminary discussion of different approaches to semi-linear waveequations we now focus our attention on (1.7) in the critical case p = 6 in di-mension n = 3, which will be fixed from now on.In the next section we present the existence result of Rauch for small data. Thenwe present an energy decay estimate and show how regularity in the radial casemay be derived.

Finally we focus on the work of Grillakis [6], whose penetratinganalysis provides the crucial insight needed to pass from the radially symmetric tothe general case and give a slightly simplified exposition. We conclude this paperwith a global existence result for certain super-critical nonlinearities and small data.3.

Rauch’s resultLet z = (x, t) denote points in space-time R3 × R. Given z0 = (x0, t0), letK(z0) =nz = (x, t); |x −x0| ≤t0 −tobe the backward light cone with vertex at z0,M(z0) =nz = (x, t); |x −x0| = t0 −toits mantle,D(t; z0) =nz = (x, t) ∈K(z0)o(t fixed)its space-like sections. If z0 = (0, 0), z0 will be omitted.

For any space-time regionQ ⊂R3 × R, T < S, we denoteQST =nz = (x, t) ∈Q; T ≤t ≤Sothe trunctated region. Hence, for instance,∂Kst = D(s) ∪D(t) ∪M st .If s = ∞or t = −∞, they will be omitted.

Given a function u on a cone K(z0), wedenote its energy density bye(u) = 12|ut|2 + |∇u|2+ 16|u|6,

SEMILINEAR WAVE EQUATIONS15and byEu; D(t; z0)=ZD(t;z0)e(u) dxits energy on the space-like section D(t; z0). Moreover, let x = y + x0 and denotedz0(u) = 12y|y|ut −∇u2+ 16|u|6the energy density of u tangent to M(z0).Finally, for x0 ∈R3 letBR(x0) =x ∈R3; |x −x0| < Rwith boundarySR(x0) =x ∈R3; |x −x0| = R.In the following, the letters c, C will denote various constants.

E0 will denote abound for the initial energy.The proof of Rauch’s existence result relies on the following inequalities of Hardy-type that also play an essential role in the work of Grillakis and this author on thelimit case p = 6. We state these estimates in a form due to Grillakis [6, Lemma2.1]).Lemma 3.1.

Suppose u ∈L6(BR) possesses a weak radial derivative ur = x·∇u|x|∈L2(BR). Then with an absolute constant C0 for all 0 ≤ρ < R the following holds:(i)ZBR\Bρ|u(x)|2|x|2dx ≤4ZBR\Bρ|ur|2 dx + 2R−1ZSR|u|2 do;(ii)ZBR|u(x)|2|x|2dx ≤C0 ZBR|ur|2 dx +ZBR|u|6 dx1/3!

;(iii)ZSR|u|4 do ≤C0 ZBR|ur|2dx1/2 ZBR|u|6 dx1/2+ZBR|u|6 dx2/3!.Proof. (i) follows upon integrating the inequality|ur|2 =1√r ∂r(√ru) −12r u2≥|u|24r2 −12r2∂∂r (r u2)over BR\Bρ.

See Grillakis [6, Lemma 2.1] for the remaining details of the proof.□Let z0 = (x0, t0) be given and suppose u is a C2-solution of (1.8) on the backwardlight cone K0(z0). As observed in §2 above, in order to prove that u can be extendedto a global solution of (1.8), it suffices to show that for any z0 as abovem0 = supK0(z0)|u|can be a priori bounded in terms of z0 and the initial data.

Clearly, we may assumethat m0 =u(z0).The first fundamental estimate towards deriving a priori bounds of this kindis the following local version of the energy inequality. For later use we refer to aslightly more general situation than described above.

16MICHAEL STRUWELemma 3.2. Let ¯z = (¯x, ¯t).

Suppose u ∈C2K0(¯z)\{¯z}solves (1.8), (1.9). Thenfor any 0 ≤t < s < ¯t there holdsEu; D(s; ¯z)+ 1√2ZMst (¯z)d¯z(u) do = E(u; D(t; ¯z)≤E0,where do denotes the surface measure on M(¯z).Proof.

Integrate the identityutt −∆u + u5ut = ddte(u) −div(∇u ut) = 0over Kst (¯z). Now let y = x −¯x and use the fact that the outward unit normal onM(¯z) is given byn =1√2 y|y|, 1.Hence the “energy flux” through M(¯z) is given byn ·−∇u ut, e(u)=1√2 12|∂tu|2 −2 y|y| · ∇u ut + |∇u|2+ 16|u|6!=1√2d¯z(u).See Rauch [12].□By Lemma 3.2, for any fixed ¯z the energy Eu; D(s; ¯z)is a monotone nonin-creasing function of s ∈[0, ¯t[ and hence converges to a limit as s ր ¯t.

It followsthat(3.1)ZMst (¯z)d¯z(u)do →0(s, t ր ¯t);however, at a rate that may depend on ¯z.Following Rauch [12] we now decompose u = v + w, where v ∈C2R3 × [0, ∞[is the unique solution of the homogeneous wave equationvtt −∆v = 0with initial data (1.9) andwtt −∆w + u5 = 0,w|t=0 = 0 = wt|t=0.In particular, at z0 = (x0, t0) we may express w via (2.3) as followsw(z0) = −14πZM0(z0)u5(x, t)t0 −t do(x, t).

SEMILINEAR WAVE EQUATIONS17Thus, and splitting integration over M T0 (z0) and MT (z0) for suitable T , weobtain(3.2)m0 =u(z0) ≤v(z0) +w(z0)≤v(z0) + m04πZMT (z0)u4t0 −t do + 14πZMT0 (z0)|u|5t0 −t do.By H¨older’s inequality and Lemma 3.2, the last term(3.3)ZMT0 (z0)|u|5t0 −t do ≤C|t0 −T |−1/2 −|t0|−1/2 ZM0(z0)|u|6 do!5/6≤C E5/60|t0 −T |−1/2 .Hence, if for some T < t0 we have(3.4)ZMT (z0)u4t0 −t do ≤2π,from (3.2) and (3.3) we can bound(3.5)m0 ≤2v(z0) + C E5/60(|t0 −T |−1/2 −|t0|−1/2)in terms of the initial data, z0, and T .Now, by H¨older’s inequalityZMT (z0)u4t0 −t do ≤ ZMT (z0)|u|2|t0 −t|2 do!1/2 ZMT (z0)|u|6 do!1/2.Let ˜u(y) = ux0 + y, t0 −|y|. Then by Lemma 3.1 we haveZMT (z0)|u|2|t0 −t|2 do =√2ZBt0−T (0)˜u(y)2|y|2dy≤CZBt0−T (0)|∇˜u|2 dy + C ZBt0−T (0)|˜u|6 dy!1/3≤CZMT (z0)dz0(u) do + C ZMT (z0)dz0(u) do!1/3≤CEu; D(T ; z0)+ E1/3u; D(T ; z0).Thus(3.6)ZMT (z0)u4t0 −t do ≤CEu; D(T ; z0)+ E2/3u; D(T ; z0).With the special choice T = 0, (3.4–6) now lead immediately to Rauch’s existenceresult:

18MICHAEL STRUWETheorem 3.3. There exists a constant ε0 > 0 such that (1.8), (1.9) for any u0 ∈C3(R3), u1 ∈C2(R3) with energyE0 =ZR3|u1|2 + |∇u0|22+ |u0|66dx < ε0admits a global C2-solution.Remark 3.4.

Estimates (3.2)–(3.6) also give the following local version of Rauch’stheorem. Let ¯z = (¯x, ¯t), ¯t > 0:If u ∈C2R3 × [0, ¯t[is a solution to (1.8), (1.9), and if(3.7)Eu; D(T ; ¯z)< ε0for some T < ¯t, then u (and its first and second derivatives) can be uniformly apriori bounded on K0(¯z) ∖{¯z} in terms of T, ¯z, and the initial data.In fact, in this case u can be extended as a solution of (1.8) to a full neighborhoodof ¯z.

Indeed, since u ∈C2R3 × [0, ¯t[, condition (3.7) will be satisfied for all points˜z = (˜x, ¯t) with ˜x close to ¯x.Finally, observe that if u (and hence its first and second derivatives) are uniformlybounded on K0(¯z) ∖{¯z}, condition (3.7) is automatically satisfied. Thus, in orderto extend u as a solution of (1.8) to a neighborhood of ¯z it suffices to establish thatlim supz0∈K(¯z)∖{¯z}z0→¯zu(z0) < ∞.By steps (3.2)–(3.6) of the proof of Rauch’s theorem then, in fact, it suffices toshow that for some T < ¯t there holds(3.8)lim supz0∈K(¯z)∖{¯z}z0→¯zZMT (z0)u4t0 −tdo ≤2π.This will be important for our next topic.4.

Large dataWe now show how the smallness assumption in Rauch’s Theorem 3.3 can be re-moved. Again remark that it suffices to consider data u0, u1 with compact support.Suppose by contradiction that u does not extend globally.Then u becomesunbounded in finite time T .Since the support of u in R3 × [0, T ] is relativelycompact there exists a “first” singular point ¯z = (¯x, ¯t), 0 < ¯t ≤T , such thatu(x, t) −→∞for some sequence x →¯x, t ր ¯t, and ¯t is minimal with this property.By Remark 3.4, in order to achieve a contradiction it suffices to establish condi-tion (3.8).Since t = 0 in the following no longer plays a distinguished role, we may shiftcoordinates so that ¯z = (0, 0) and henceforth assume that u is a C2-solution of (1.8)

SEMILINEAR WAVE EQUATIONS19on R3 × [t1, 0[ for some t1 < 0. As customary, the Landau symbol “o(1) as r →0”will denote error terms depending on a parameter r that tend to 0 as r →0.Observe that (1.8) and E are invariant under scalingR 7→uR(x, t) = R1/2u(Rx, Rt),for any R > 0.Following Struwe [18, Lemma 2.3] we use the testing function t ut + x · ∇u + uto derive the following identity(4.1) 0 =utt −∆u + u5t ut + x · ∇u + u= ddtt Q0 + utu−div(t P0) + R0,whereP0 = xt 12|ut|2 −12|∇u|2 −16|u|6+ut + xt · ∇u + ut∇u,Q0 = 12|ut|2 + 12|∇u|2 + 16|u|6 +xt · ∇uut ≥0in Kt1,R0 = 13|u|6 ≥0.Note that the multiplier tut + x · ∇u + u is related to the generator of the scaledfamily uR.

As in Grillakis [6, (2.2)], we may rewrite (4.1) in the form(4.2)0 = t(ddtQ0 + utut+ u22t2−div P0)+(Q0 + u2t2 + R0).If we integrate (4.1) over a truncated cone Kst , integrals involving utu will appear.Using the functiont2 + |x|2ut + 2t x · ∇u + 2tu as a further multiplier, Grillakissucceeds in showing that1|t|ZD(t)utu dx ≤o(1) →0(t ր 0).With little more extra work this additional multiplier can be avoided.As a first step, we obtainLemma 4.1 [18, Lemma 3.2]. There exists a sequence of numbers tℓր 0 such that1|tl|ZD(tl)utu dx ≤o(1),where o(1) →0 as l →∞.For completeness we present the proof.Proof.

Consider ul(x, t) = 2−l/2u(2−lx, 2−lt), l ∈N, satisfying (1.8) withEul; D(t)= Eu; D(2−lt)≤E0;moreover, (3.1) translates into the conditiond(u) := d0(u)(4.3)ZMt1d(ul) do →0

20MICHAEL STRUWEas l →∞. First, suppose thatZD(t1)u2l dx →0(l →∞).Then let tl = 2−ℓt1 and estimate1|tl|ZD(tl)utu dx ≤ ZD(tl)|ut|2 dx!1/2 1|tl|2ZD(tal)u2 dx!1/2≤2Eu; D(tl)1/2 1|t1|2ZD(t1)u2l dx!1/2→0(l →∞)(4.4)to achieve the claim.Otherwise, there exist C1 > 0 and a sequence Λ of numbers l →∞such thatlim infl→∞, l∈ΛZD(t1)u2l dx ≥C1.For any s ∈[t1, 0[, by H¨older’s inequality(4.5)ZD(s)u2l dx ≤4π3 |s|32/3 ZD(s)|ul|6 dx!1/3≤C E1/30s2.Choose s = s1 < 0 such that the latter is ≤C1.

Then by (4.3) we have2ZKs1t1(ul)tul dz =ZKs1t1ddt|ul|2 dx=ZD(s1)|ul|2 dx −ZD(t1)|ul|2 dx + 1√2ZMs1t1|ul|2 do≤o(1) →0(l →∞, l ∈Λ).We conclude that for suitable numbers sl ∈[t1, s1], tl = 2−ℓsl, l ∈Λ, we have2|tl|ZD(tl)utu dx =2|sℓ|ZD(sl)(ul)tul dx ≤o(1) →0(l →∞, l ∈Λ).Relabelling, we obtain a sequence (tl)l∈N, as desired.□Lemma 4.2 [18, Lemma 2.2]. For any l ∈N there holds13|tl|ZKtl|u|6 dz +ZD(tl)12|ut|2 + 12|∇u|2 + utxt · ∇u+ 16|u|6dx≤o(1) →0(l →∞).Again we give the proof for completeness.

SEMILINEAR WAVE EQUATIONS21Proof. For s ∈[tl, 0[ integrate (4.1) over Kstl to obtain0 =ZD(s)s Q0 + utudx + 1√2ZMstlt Q0 + utu −x · P0do+ |tl|ZD(tl)Q0 dx +ZKstlR0 dx −ZD(tl)utu dx.Now, Q0 is dominated by the energy density e(u).

Thus, and using H¨older’s in-equality as in (4.4), (4.5), the first term is of order |s| and hence vanishes as s ր 0.Moreover, on Mtl we havet Q0 + utu −x · P0= |t||∇u|2 −x|x| · ∇u2+ 13|u|6 −u(t ut + x · ∇u)t2≤|tl|3d0(u) + |u|2t2.Hence by (3.1) and Hardy’s inequality Lemma 3.1 the second term is of ordero(1)|tl|, where o(1) →0 as l →∞. Thus, by Lemma 4.1 we have1|tl|ZKstℓR0 dz +ZD(tl)Q0 dx ≤1|tl|ZD(tl)utu dx + o(1) ≤o(1) →0(l →∞),which is the desired conclusion.□Now we use (4.1) in its equivalent form (4.2) to derive a stronger version ofLemma 4.1.Lemma 4.3.

There exists a sequence of numbers ¯tl ր 0 such that the conclusionof Lemma 4.1 holds for (¯tl) while in addition we have2 ≤¯tl/¯tl+1 ≤4for all l ∈N.Proof. First observe that by H¨older’s inequality and Lemma 4.2 for any m ∈N wehaveZD(tm)|u|2|t|2 dx ≤C ZD(tm)|u|6 dx!1/3≤C ZD(tm)Q0 dx!1/3−→0(m →∞),where (tm) is determined in Lemma 4.1.

Divide (4.2) by t and integrate over thecone Ktmtlfor m ≥l to obtainZD(tl)Q0 + utut+ u22t2dx +ZKtmtlQ0|t| + |u|2|t|3 + R0|t|dz=ZD(tm)Q0 + utut+ u22t2dx +ZMtmtlQ0 + utut+ u22t2 −xt · P0do.

22MICHAEL STRUWEBy the preceding remark the first term on the right vanishes as we let m →∞,while by (3.1) the integral over M tmtℓbecomes arbitrarily small as m ≥l →∞.Finally, by Lemma 4.1, we haveZD(tl)ututdx = −1|tl|Zutu dx ≥o(1) →0(l →∞).All remaining terms being nonnegative, we thus obtain the estimateZKtl|u|2|t|3 dz =Z 0tl 1|t|ZD(t)|u|2|t|2 dx!dt ≤o(1) →0(l →∞).Hence for any ¯t ∈[tl/2, 0[ there also holdso(1) ≥1¯tZ ¯t2¯t ZD(t)|u|2|t|2 dx!dt ≥inf2¯t≤t≤¯tZD(t)|u|2|t|2 dt,where o(1) →0 if l →∞.To construct the sequence (¯tl), now choose ¯t1 = t1 and proceed by induction.Suppose ¯tl, l = 1, ..., L, have been defined already. Let ¯tL+1 ∈h¯tL2 , ¯tL4hbe chosensuch thatZD¯tL+1 |u|2|t|2 dx ≤2inf¯tL2 ≤t≤¯tL4ZD(t)|u|2|t|2 dx.Clearly, this procedure yields a sequence (¯tl) such that 2 ≤¯tl/¯tl+1 ≤4 for all l andZD(¯tl)|u|2|t|2 dx −→0(l →∞).By (4.4) then1|¯tl|ZD(¯tℓ)utu dx −→0(l →∞),concluding the proof.□To simplify notation, we will assume that tl = ¯tl for all l, initially.The radial case.

At this point we can indicate how the decay estimate Lemma4.2 and Lemma 4.3 imply regularity of solutions in the radial case. First observethat for radially symmetric data u0(x) = u0|x|, etc., the unique local C2-solutionu to (1.8), (1.9) again is radially symmetric, that is, u(x, t) = u|x|, t.Note that this implies that blow-up can only occur on the line x = 0.

Indeed, ifu is regular on K0(¯z)\{¯z} and blows up at ¯z = (¯z, ¯t), the same will be true for anypoint z = (x, ¯t) with |x| = |¯x| = ¯r. Now, if ¯x ̸= 0, given any K ∈N we can choosepoints xk ∈R3 with |xk| = ¯r, 1 ≤k ≤K, and T ∈[0, ¯t[ such thatD(T ; zk) ∩D(T ; zl) = ∅for all k ̸= l, where zk = (xk, T ), k = 1, ..., K. Moreover, by Remark 3.4Eu; D(T ; zk)≥ε0 > 0

SEMILINEAR WAVE EQUATIONS23Figure 1. The energy at the basis of each cone is ≥ε0.for all k, while by Lemma 3.2Kε0 ≤KXk=1Eu; D(T ; zk)= E u;K[k=1D(T ; zk)!≤Eu; DT ; (0, ¯r + ¯t)≤Eu; D0; (0, ¯r + ¯t)≤E0,independently of K. Thus, blow-up may first occur on the line x = 0, only.

(SeeFigure 1. )Let blow-up occur at (0, ¯t).

Shifting time by ¯t then we may assume that u(x, t) =u|x|, tis regular on R3 × [t1, 0[ and blows up at the origin. As a second step weestimate the speed of blow-up.

Again observe that (1.8) is invariant under scalingu 7−→uR(x, t) = R1/2u(R x, R t).This suggests that u(z) ∼|z|−1/2. In fact, the following result holds.

24MICHAEL STRUWEFigure 2. Overlap of the cones Ktl−L(zjk).Lemma 4.4 [18; Lemma 3.3].4ε1 := lim supz=(x,t)→0z∈Kt1u(z) · |z|1/2> 0.The proof of Lemma 4.4 is rather involved and will not be presented here.We can now conclude the regularity proof.

Let ¯zk = (¯xk, ¯sk) ∈Kt1 satisfyu(¯zk) =supz∈Kt1(¯zk)u(z) ≥2ε1|¯zk|−12 .Choose a sequence l = l(k) →∞(k →∞) such thattl+1 ≥¯sk ≥tl,with (tl) as in Lemma 4.3. By (3.2)–(3.3) we may fix L ∈N independent of k suchthat for large k there holdsu(¯zk) 1 −14πZMtl−L (¯zk)u4¯sk −t do!≤C + C|¯sk −tl−L|−1/2 −|¯sk −t1|−1/2E5/60≤C + C 2−L/2|¯sk|−1/2E5/60≤ε1|¯zk|−1/2,where E0 = Eu; D(t1, 0)is the initial energy.Thus, by choice of ¯zk and (3.6) we conclude thatEu; D(tl−L; ¯zk)≥ε0for all k.Given J ∈N, for each k ∈N choose J points xjk, j = 1, ..., J, equi-distributed onthe sphere |xjk| = |¯xk|.

Let zjk = (xjk, ¯sk). Note that there exists δ = δ(J, L) > 0

SEMILINEAR WAVE EQUATIONS25such that (x, tl−L) ∈D(tl−L; zik)∩D(tl−L; zjk), i ̸= j, implies that |x| ≤(1−δ)tl−L. (See Figure 2.

)Hence by Lemma 4.2 we haveJε0 ≤JXj=1Eu; D(tl−L; zjk)≤Eu; D(tl−L; 0)+Xi̸=jEu; D(tl−L; zik) ∩D(tl−L; zjk)≤E0 + C(J, δ)ZD(tl−L)Q0 dx ≤E0 + o(1),where o(1) →0 as k →∞for any fixed J. Choosing J large, for sufficiently largek ∈N we thus obtain a contradiction.

Hence, u is uniformly bounded on Kt1 andthe proof in the radially symmetric case is complete.General data. Finally, we present Grillakis’ work on the general case.

The key ob-servation is that the decay Lemma 4.2 suffices to establish (3.8), directly. However,this is not at all easy to see.Fix z0 = (x0, t0) ∈K\{0} arbitrarily.

Denote y = x −x0, ˆy =y|y|, ˆx =x|x|.Divide (4.2) by t and for s > t0 integrate over Kstl\K(z0) to obtain the relation0 =ZD(s)Q0 + utut+ u22t2dx −ZD(tl)\D(tl;z0)Q0 + utut+ u22t2dx+ 1√2ZMstlQ0 + utut+ u22t2 −ˆx · Pdo−1√2ZMtl (z0)Q0 + utut+ u22t2 −ˆy · Pdo+ZKstl \K(z0) R0 + Q0 + u22t2t!dz = I + · · · + V.(See Figure 3.)

26MICHAEL STRUWEFigure 3By H¨older’s inequality (4.4), (4.5) and Lemma 4.2 the first term I →0 if wechoose s = tk with k →∞. Similarly, II →0 if l →∞.

By (3.1) also III →0 asl →∞. Finally V ≤0.

Thus we obtain the estimate for any z0 ∈K\{0},(4.6)ZMtl (z0)Q0 + utut+ u22t2 −ˆy · Pdo ≤o(1) →0(l →∞),with error term o(1) independent of z0.By a beautiful geometric reasoning, Grillakis [6] now proceeds to bound (3.8) interms of (4.6) of (4.6). Let r = |x|; then we may rewriteA : = Q0 + utut+ u22t2 −ˆy · P=1 −ˆx · ˆy rt12|ut|2 +1 + ˆx · ˆyrt 12|∇u|2 + 16|u|6+ 1tut −ˆy · ∇uu + rt utˆx · ∇u −utˆy · ∇u −rtˆx · ∇uˆy · ∇u+ u22t2 .Introducing uσ = ˆy · ∇u, α = ˆx −ˆy(ˆy · ˆx), |α|uα = α · ∇u, Ωu = ∇u −ˆyuσ, thisexpression becomes=1 −ˆx · ˆyrt12(ut −uσ)2 +1 + ˆx · ˆy rt 12|Ωu|2 + 16|u|6+ rt |α|(ut −uσ)uα + ut (ut −uσ) + u22t2 .Now let ˆx · ˆy = cos δ, |α| = sin δ and for brevity denote1√2(ut −uσ) = uρ.

(SeeFigure 4 on p. 78.) Then the aboveA =1 −rt cos δ|uρ|2 +1 + rt cos δ 12|Ωu|2 + 16|u|6+√2rt | sin δ|uρuα +√2uρut+ u22t2= : A0 +√2uρut+ u22t2 .

(4.7)

SEMILINEAR WAVE EQUATIONS27Figure 4Note that if we estimate |uα| ≤|Ωu|, in particular, we haveA0 ≥1 −rt cos δ|uρ|2 +1 + rt cos δ 12|uα|2 + 16|u|6+√2rt | sin δ|uρuα≥1 + rt|uρ|2 + 12uα|2 + 16|u|6−r2t√2√1 + cos δuρ −√1 −cos δuα2≥0(4.8)on Mtl(z0).Now for any ε > 0 there is a constant C = C(ε) such that for any z0 ∈K andany z ∈M Ct0(z0) we may estimate−√2rt | sin δ| ≤ε,−rt cos δ ≥12.In fact, for z = (x, t) ∈M Ct0(z0) we have|x| −|y| ≤|y −x| = |x0| ≤|t0| ≤|t −t0|C −1 =|y|C −1.Henceˆx · ˆy = cos δ = x|x| · y|y| ≥1 −y|y| −x|x| ≥1 −2|x0||y| ≥1 −2C −1

28MICHAEL STRUWEwhile1 ≥−rt =|y||t −t0| · |t −t0||t|· |x||y| ≥1 −1C 1 −1C −1.This yields the following estimate.Lemma 4.5. For any ε > 0, any z0 ∈K, letting C = C(ε) be determined as above,if tk ≤C t0 we haveZMtktl (z0)A do ≥12ZMtktl (z0)|uρ|2 do −εE0.Proof.

Estimate√2uρut ≤|uρ|2 + u22t2 .Hence by (4.7) and our choice of C(ε), for z ∈M C t0tl(z0) we haveA ≥12|uρ|2 −ε|uρuα| ≥12|uρ|2 −1√2 ∈dz0(u),which in view of Lemma 3.2 proves the claim.□Observe that uρ may be interpreted as a tangent derivative along M(z0). Infact, let Φ be the map(4.9)Φ : y 7→(x0 + y, t0 −|y|)and let(4.10)v(y) = uΦ(y),whenever the latter is defined.

Then the radial derivative vs of v is given by(4.11)vs = y · ∇v|y|= uσ −ut = −√2uρ.Lemma 4.6 (Grillakis [6, (2.23)]). For any z0 ∈K and any C ≥0 there holdsZM(1+C)t0 (z0)uρutdo ≥1 + ln(1 + C)· o(1)where o(1) →0 if (1 + C)t0 ≥tl and l →∞.Proof.

Introducing y as new variable via (4.9), (4.10), we haveZM(1+C)t0 (z0)uρutdo =ZBC|t0|vsv|y| −t0dy =ZS1 Z C|t0|0vsvs −t0s2 ds!do.

SEMILINEAR WAVE EQUATIONS29Upon integrating by parts, this givesZS1 Z C|t0|0∂s(v2/2)s −t0s2 ds!do=ZS1Z C|t0|0−v2ss −t0+v2s22(s −t0)2ds do +12(1 + C)|t0|ZSC|t0|v2 do≥−ZBC|t0|v2|y||y| −t0 dy = −1√2ZM(1+C)t0 (z0)u2t(t −t0) do(x, t)= −Z t0(1+C)t01|t| 1|t −t0|Z∂D(t;z0)u2 do(x)!dt.Now by Hardy’s inequality Lemma 3.1. (iii) we have 1|t −t0|Z∂D(t;z0)u2 do!2≤CZ∂D(t;z0)u4 do≤C ZD(t;z0)|∇u|2 dx!1/2+ ZD(t;z0)|u|6 dx!1/6· ZD(t;z0)|u|6 dx!1/2≤C(E0) ZD(t;z0)|u|6 dx!1/2.HenceZM(1+C)t0 (z0)uρutdo ≥−CZ t0(1+C)t01|t| ZD(t)|u|6 dx!1/4dt,with C depending on E0 only.

By Lemma 4.2 the latter can be controlled as follows.Let k, K ∈N be determined such thattk ≤(1 + C)t0 < tk+1 ≤tK ≤t0 < tK+1.Note that by Lemma 4.31 + C ≥tk+1tK≥2K−(k+1),whenceK −k ≤1 + log2(1 + C).EstimateZ t0(1+C)t01|t| ZD(t)|u|6 dx!1/4dt ≤KXi=kZ ti+1ti1|t| ZD(t)|u|6 dx!1/4dt.By H¨older’s inequality, this is≤CKXi=kti −ti+13/4ti+1 ZKti+1ti|u|6 dz!1/4

30MICHAEL STRUWEFigure 5and by Lemma 4.3≤CKXi=k 1|ti|ZKti|u|6 dz!1/4.Finally, use Lemma 4.2 to see that this is≤(K −k + 1)o(1) ≦1 + ln(1 + C)o(1)where o(1) →if (1 + C)t0 ≥tl and l →∞.□Combining Lemmas 4.5 and 4.6 it follows that for any ε > 0, if we choosetk ≤C(ε)t0 < tk+1, we can estimate(4.12)o(1) ≥ZMtl (z0)A do≥12ZMtktl (z0)|uρ|2 do −εE0 +ZMtk (z0)A0 do −o(1)1 + ln1 + C(ε),where o(1) →0 as l →∞. To estimate A0 on Mtk(z0) now introduce the new angleδ0, where |x0| = r0, ˆx0 = x0r0 , ˆx0 · ˆy = cos δ0.

(See Figure 5.) Again y = x −x0,and |y| = σ = t0 −t.

With this notationr cos δ = ˆx · ˆyr = x · ˆy = y · ˆy + x0 · ˆy = σ + r0 cos δ0,

SEMILINEAR WAVE EQUATIONS31| sin δ| = |α| =x −(x · ˆy)ˆyr =x0 −(x0 · ˆy)ˆyr = r0r | sin δ0|.Hence, by (4.7),(4.13)A0 =1 −σt −r0t cos δ0|uρ|2 +1 + σt + r0t cos δ0 12|Ωu|2 + 16|u|6+√2 r0t | sin δ0|uρuα.Estimating |Ωu| ≥|uα| as before, this is(4.14)≥2 −t0 −r0t|uρ|2 −r02t√2p1 + cos δ0uρ −p1 −cos δ0uα2+ t0t1 + r0t012|uα|2 + t0t1 + r0t0cos δ016|u|6.Note that all the latter terms are nonnegative for z ∈M(z0), z0 ∈K. By (4.14), andsince r0 ≤|t0|, for t ≤2t0 we have A0 ≥|uρ|2.

Moreover, given 0 < ε < 1, z0 ∈K,let tm ≤2t0 < tm+1 and setΓ = Γ(ε; z0) =nz ∈Mtm(z0); |δ0| ≤ε1/4o∆= ∆(ε; z0) = Mtm(z0)\Γ.Note that by (4.13) on Γ we can estimateA0 ≥|uρ|2 −√2ε1/4|uρuα|≥|uρ|2 −√2ε1/4dz0(u),while by (4.14) on ∆we haveA0 ≥t0t1 + r0t0cos δ016|u|6 ≥18 1 −1 −ε1/22+ ε!16|u|6≥ε1/296 |u|6 −εdz0(u).Combining with (4.12) and Lemma 4.5, thus we obtainZΓ|uρ|2do ≤ZMtk (z0)A0do +√2ε1/4E0≤ε +√2ε1/4E0 + o(1)1 + ln1 + C(ε),(4.15)(4.16)ε1/296Z∆|u|6 do ≤ZMtk (z0)A0do + εE0 ≤2εE0 + o(1)1 + ln1 + C(ε),(4.17)ZMtmtl(z0)|uρ|2 do ≤ZMtmtk (z0)A0 do +ZMtktl (z0)|uρ|2 do ≤2εE0 + o(1) ln1 + C(ε),

32MICHAEL STRUWEwhere o(1) →0 as l →∞. (We may assume tl ≤tk ≤tm.

)Proof of Theorem 1.1. Given ε > 0, we split the integral in (3.8) and use H¨older’sinequality as followsZMtl (z0)|u|4|t −t0| do ≤ZΓ+ · · · +Z∆+ · · · +ZMtmtl (z0)+ · · ·≤ZΓ|u|2|t −t0|2 do1/2 ZΓ|u|6 do1/2+Z∆|u|2|t −t0|2 do1/2 Z∆|u|6 do1/2+ ZMtmtl (z0)|u|2|t −t0|2 do!1/2 ZMtmtl (z0)|u|6 do!1/2.By Lemma 3.1.

(ii) and Lemma 3.2 this can be further estimated≤p6E0ZΓ|u|2|t −t0|2 do1/2−C(E0)Z∆|u|6 do1/2+p6E0 ZMtmtl (z0)|u|2|t −t0|2 do!1/2.By Lemma 3.1. (i) and (4.15)ZΓ|u|2|t −t0|2 do ≤4ZΓ|uρ|2 do + 2|tm −t0|−1Z∂D(tm; z0)|u|2 do≤4ε +√2ε1/4E0 + o(1)1 + ln1 + C(ε)+ C Z∂D(tm; z0)|u|4 do!1/2.By Lemma 3.1.

(iii) and Lemma 4.2 the latterZ∂D(tm,z0)|u|4 do ≤C ZD(tm)|∇u|2 dx!1/2+ ZD(tm)|u|6 dx!1/6 ZD(tm)|u|6 dx!1/2≤C(E0)o(1),where o(1) →0 as m ≥l tend to infinity. Similarly, by Lemma 3.1.

(i), (iii), Lemma4.2 and (4.17)ZMtmtl(z0)|u|2|t −t0|2 do ≤4ZMtmtl (z0)|uρ|2 do + 2|tl −t0|−1Z∂D(tl;z0)|u|2 do≤εE0 + o(1)C(E0) + ln1 + C(ε).

SEMILINEAR WAVE EQUATIONS33Finally, by (4.16)Z∆|u|6 do ≤192ε1/2E0 + o(1)ε−1/21 + ln1 + C(ε).Hence, if we first choose ε > 0 sufficiently small and then choose l ∈N sufficientlylarge, the integralZMtl (z0)u4|t −t0| docan be made as small as we please, uniformly in z0 ∈Kt1.□Remark. Since all error estimates are based on the qualitative statement (3.1), noa priori bounds for the solution u on a cone K, depending only on u0, u1, and K,are obtained.5.

A remark on the super-critical caseWe add an observation on the super-critical case. Consider for simplicity theequation(5.1)utt −∆u + u5 + u|u|p−2 = 0in R3 × [0, ∞[with initial data (1.2).

(5.1) may be approximated by equations(5.2)utt −∆u + u51 + min|u|p−6, kp−6= 0.By the preceding, (5.2) admits global C2-solutions u(k); moreover, as in Rauch’sTheorem 3.3 we may decomposeu(k)(z0) = u(0)(z0) + v(k)(z0),where u(0) solves the homogenous wave equation with initial data u0, u1. Now,for suitable initial data, we obtain a uniform boundu(0)(z) ≤m0for all z ∈R3 × [0, ∞[; for instance, if u0, u1 have compact support.

By (2.3), moreover, ifu(k)(z0) =supz∈K0(z0)u(k)(z) = mk,we may estimatemk =u(k)(z0) ≤u(0)(z0) +v(k)(z0)≤m0 + mk4πZM0(z0)|u(k)|41 + min|u(k)|p−6, kp−6t0 −tdo≤m0 + mk1 + kp−64πZM0(z0)|u(k)|4t0 −t do≤m0 + C mkkp−6Eu(0)+ E2/3u(0)< 2m0,if k = 2m0 and if Eu(0)is sufficiently small, depending on m0, that is, on u0 andu1. Thus u = u(k) solves (5.1).In particular, we obtain the following perturbation result:

34MICHAEL STRUWETheorem 5.1. Suppose u0 ∈C3(R3), u1 ∈C2(R3) have finite energyZR3|u1|2 + |∇u0|22+ |u0|66dx < ∞and suppose the solution u(0) to the homogeneous wave equation with initial datau0, u1 is uniformly bounded.

Then there exists ε0 > 0 such that for all |ε| < ε0 theinitial value problem for (5.1) with data εu0, εu1 admits a global C2-solution.However, “in the large” the super-critical case appears to be completely open.References1. P. Brenner, On Lp −Lp′ estimates for the wave equation, Math.

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