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On the critical semilinear wave equation outside convex obstacles


Authors: Hart F. Smith and Christopher D. Sogge
Journal: J. Amer. Math. Soc. 8 (1995), 879-916
MSC: Primary 35L70; Secondary 58G16, 58G20, 58G25
DOI: https://doi.org/10.1090/S0894-0347-1995-1308407-1
MathSciNet review: 1308407
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  • [1] M. Farris, Egorov's theorem on a manifold with diffractive boundary, Comm. Partial Differential Equations 6 (1981), 651-688. MR 617785 (82m:58050)
  • [2] F. G. Friedlander, The wave front set of the solution of a simple initial boundary value problem with glancing rays, Proc. Cambridge Philos. Soc. 79 (1976), 145-159. MR 0393861 (52:14669)
  • [3] D. Grieser, $ {L^p}$ bounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries, Thesis, UCLA, 1992.
  • [4] M. G. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Ann. of Math. (2) 132 (1990), 485-509. MR 1078267 (92c:35080)
  • [5] -, Regularity for the wave equation with a critical nonlinearity, Comm. Pure Appl. Math. 45 (1992), 749-774. MR 1162370 (93e:35073)
  • [6] L. Hörmander, Fourier integrals I, Acta Math. 127 (1971), 79-183.
  • [7] -, The analysis of linear partial differential operators, Vols. I-IV, Springer-Verlag, Berlin, 1983, 1985.
  • [8] -, Non-linear hyperbolic differential equations, Lund lecture notes, 1988 .
  • [9] L. Kapitanski, Cauchy problem for a semilinear wave equation II, J. Soviet Math. 62 (1992), 2746-2776; III, 2619-2645. MR 1097579 (92d:35200)
  • [10] -, Some generalizations of the Strichartz-Brenner inequality, Leningrad Math. J. 1 (1990), 693-726. MR 1015129 (90h:46063)
  • [11] S. Klainerman and M. Machedon, The null condition and local existence for nonlinear waves, Comm. Pure Appl. Math. 46 (1993), 1221-1268. MR 1231427 (94h:35137)
  • [12] H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. (to appear). MR 1335386 (96i:35087)
  • [13] R. Melrose, Microlocal parametrices for diffractive boundary value problems, Duke Math. J. 42 (1975), 605-635. MR 0517101 (58:24409)
  • [14] -, Equivalence of glancing hypersurfaces, Invent. Math. 37 (1976), 165-191. MR 0436225 (55:9173)
  • [15] -, Transformation of boundary value problems, Acta Math. 147 (1981), 149-236.
  • [16] R. Melrose and M. Taylor, Near peak scattering and the corrected Kirchoff approximation for a convex obstacle, Adv. Math. 55 (1985), 242-315. MR 778964 (86m:35095)
  • [17] -, The radiation pattern of a diffractive wave near the shadow boundary, Comm. Partial Differential Equations 11 (1985), 599-672. MR 837278 (87i:35109)
  • [18] -, Boundary problems for the wave equation with grazing and gliding rays, manuscript.
  • [19] G. Mockenhaupt, A. Seeger, and C. D. Sogge, Local smoothing of Fourier integrals and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), 65-130. MR 1168960 (93h:58150)
  • [20] C. S. Morawetz, Energy decay for star-shaped obstacles, Scattering Theory (P. D. Lax and R. S. Phillips, eds.), Academic Press, San Diego, 1989, pp. 261-264.
  • [21] H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordan equations, Math. Z. 185 (1984), 261-270. MR 731347 (85h:35165)
  • [22] J. Rauch, The $ {u^5}$-Klein-Gordan equation, Nonlinear PDE's and their Applications, Pitman Res. Notes Math. Ser., vol. 53, Longman Sci. Tech., Harlow, 1976, pp. 335-364. MR 631403 (83a:35066)
  • [23] J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann of Math. (2) 138 (1993), 503-518. MR 1247991 (95f:35164)
  • [24] A. Seeger and C. D. Sogge, Bounds for eigenfunctions of differential operators, Indiana Math. J. 38 (1989), 669-682. MR 1017329 (91f:58097)
  • [25] H. F. Smith and C. D. Sogge, $ {L^p}$ regularity for the wave equation with strictly convex obstacles, Duke Math. J. 73 (1994), 97-155. MR 1257279 (95c:35048)
  • [26] -, On Strichartz and eigenfunction estimates for low regularity metrics, Math. Research Letters 1 (1994), 729-737. MR 1306017 (95h:35156)
  • [27] C. D. Sogge, Concerning the $ {L^p}$ norm of spectral clusters for second order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), 123-134. MR 930395 (89d:35131)
  • [28] -, Fourier integrals in classical analysis, Cambridge Univ. Press, Cambridge and New York, 1993. MR 1205579 (94c:35178)
  • [29] E. M. Stein, Singular integrals and differentiablity properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
  • [30] -, Harmonic analysis real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)
  • [31] R. Strichartz, A priori estimates for the wave equation and some applications, J. Funct. Anal. 5 (1970), 218-235. MR 0257581 (41:2231)
  • [32] -, Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation, Duke Math. J. 44 (1977), 705-714. MR 0512086 (58:23577)
  • [33] M. Struwe, Globally regular solutions to the $ {u^5}$ Klein-Gordon equation, Ann. Sci. Norm. Sup. Pisa 15 (1988), 495-513. MR 1015805 (90j:35142)
  • [34] -, Semilinear wave equations, Bull. Amer. Math. Soc. (N.S.) 26.(1992), 53-85.
  • [35] M. Taylor, Grazing rays and reflection of singularities of solutions to wave equations, Comm. Pure Appl. Math. 29 (1976), 1-38. MR 0397175 (53:1035)
  • [36] -, Diffraction effects in the scattering of waves, Singularities in Boundary Value Problems, Reidel, Dordrecht, 1981, pp. 271-316. MR 617235 (84h:78016)
  • [37] M. Zworski, High frequency scattering by a convex obstacle, Duke Math. J. 61 (1990), 545-634. MR 1074308 (92c:35070)

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DOI: https://doi.org/10.1090/S0894-0347-1995-1308407-1
Article copyright: © Copyright 1995 American Mathematical Society

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