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Journal of the American Mathematical Society
ISSN 1088-6834(e) ISSN 0894-0347(p)

     

On the critical semilinear wave equation outside convex obstacles

Author(s): Hart F. Smith; Christopher D. Sogge
Journal: J. Amer. Math. Soc. 8 (1995), 879-916.
MSC: Primary 35L70; Secondary 58G16, 58G20, 58G25
MathSciNet review: 1308407
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References:

[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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Additional Information:

DOI: 10.1090/S0894-0347-1995-1308407-1
PII: S0894-0347-1995-1308407-1
Copyright of article: Copyright 1995, American Mathematical Society




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