# zbMATH — the first resource for mathematics

Global existence for energy critical waves in 3-D domains. (English) Zbl 1204.35119
The authors prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on $$H^1_0(\Omega) \times L^2( \Omega)$$ for any smooth (compact) domain $$\Omega \subset \mathbb R^3$$. The main ingredient in the proof is an $$L^5$$ spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.

##### MSC:
 35L71 Second-order semilinear hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations
Full Text:
##### References:
 [1] Ramona Anton. Strichartz inequalities for lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, 2005. Preprint, arXiv:math.AP/0512639. · Zbl 1157.35100 [2] N. Burq, P. Gérard, and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), no. 3, 569 – 605. · Zbl 1067.58027 [3] Jacques Chazarain and Alain Piriou, Introduction à la théorie des équations aux dérivées partielles linéaires, Gauthier-Villars, Paris, 1981 (French). · Zbl 0446.35001 [4] Jacques Chazarain and Alain Piriou, Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. · Zbl 0487.35002 [5] Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409 – 425. · Zbl 0974.47025 [6] Manoussos G. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. (2) 132 (1990), no. 3, 485 – 509. · Zbl 0736.35067 [7] Sergiu Klainerman and Matei Machedon, Remark on Strichartz-type inequalities, Internat. Math. Res. Notices 5 (1996), 201 – 220. With appendices by Jean Bourgain and Daniel Tataru. · Zbl 0853.35062 [8] Gilles Lebeau. Estimation de dispersion pour les ondes dans un convexe. In Journées “Équations aux Dérivées Partielles” (Evian, 2006), 2006. [9] Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), no. 1, 65 – 130. · Zbl 0776.58037 [10] Jeffrey Rauch, I. The \?$$^{5}$$ Klein-Gordon equation. II. Anomalous singularities for semilinear wave equations, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. I (Paris, 1978/1979) Res. Notes in Math., vol. 53, Pitman, Boston, Mass.-London, 1981, pp. 335 – 364. [11] Jalal Shatah and Michael Struwe, Regularity results for nonlinear wave equations, Ann. of Math. (2) 138 (1993), no. 3, 503 – 518. · Zbl 0836.35096 [12] Jalal Shatah and Michael Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices 7 (1994), 303ff., approx. 7 pp., issn=1073-7928, review=\MR{1283026}, doi=10.1155/S1073792894000346,. · Zbl 0830.35086 [13] Hart F. Smith and Christopher D. Sogge, On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc. 8 (1995), no. 4, 879 – 916. · Zbl 0860.35081 [14] Hart F. Smith and Christopher D. Sogge, On the \?^{\?} norm of spectral clusters for compact manifolds with boundary, Acta Math. 198 (2007), no. 1, 107 – 153. · Zbl 1189.58017 [15] Michael Struwe, Globally regular solutions to the \?$$^{5}$$ Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 3, 495 – 513 (1989). · Zbl 0728.35072 [16] Terence Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, J. Differential Equations 189 (2003), no. 2, 366 – 382. · Zbl 1017.81037 [17] Daniel Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc. 15 (2002), no. 2, 419 – 442. · Zbl 0990.35027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.