MR2761087 Soichiro KATAYAMA and Hideo KUBO Soichiro KATAYAMA and Hideo KUBO The Rate of Convergence to the Asymptotics for the Wave Equation in an Exterior Domain Funkcialaj Ekvacioj. Serio Internacia 53 2010 331--358 http://fe.math.kobe-u.ac.jp/FE/FullPapers/53-3/53_331.pdf http://www.ams.org/mathscinet-getitem?mr=MR2761087 In this paper we consider the mixed problem for the wave equation exterior to a non-trapping obstacle in odd space dimensions. We derive a rate of the convergence of the solution for the mixed problem to its asymptotic profile, which is written as a solution for the Cauchy problem. As a by-product, we are able to find out the radiation field of solutions to the mixed problem in terms of the scattering data. Wave equation, Exterior problem, Asymptotic behavior, Convergence rate. 35L20; 35B40. 53-331 2010 The Rate of Convergence to the Asymptotics for the Wave Equation in an Exterior Domain Soichiro KATAYAMA and Hideo KUBO Soichiro KATAYAMA and Hideo KUBO 1 Courant, R.; Hilbert, D. Methods of Mathematical Physics II Interscience, New York 1962 MR0140802
2 Friedlander, F. G. On the radiation field of pulse solutions of the wave equation Proc. Roy. Soc. Ser. A 269 1962 53-65 MR0142888
3 Friedlander, F. G. On the radiation field of pulse solutions of the wave equation. II Proc. Roy. Soc. Ser. A 279 1964 386-394 MR0164132
4 Friedlander, F. G. On the radiation field of pulse solutions of the wave equation. III Proc. Roy. Soc. Ser. A 299 1967 264-278 MR0226218
5 Friedlander, F. G. Radiation fields and hyperbolic scattering theory Math. Proc. Cambridge Philos. Soc. 88 1980 483-515 MR0583989
6 Hörmander, L. Lectures on Nonlinear Hyperbolic Differential Equations Springer-Verlag, Berlin 1997 MR1466700 7 Ikawa, M. Hyperbolic Partial Differential Equations and Wave Phenomena Translations of Mathematical Monographs 189, Iwanami Series in Modern Mathematics, AMS, Providence, RI 2000 MR1756774
8 John, F. Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data Comm. Pure Appl. Math. 40 1987 79-109 MR0865358
9 Lax, P. D.; Phillips, R. S. Scattering Theory Revised Edition, Academic Press, New York 1989 MR1037774
10 Melrose, R. B. Singularities and energy decay in acoustical scattering Duke Math. J. 46 1979 43-59 MR0523601
11 Shibata, Y.; Tsutsumi, Y. Global existence theorem for nonlinear wave equation in exterior domain "Recent topics in nonlinear PDE (Hiroshima, 1983)", North-Holland Math. Stud., 98, North-Holland, Amsterdam 1984 155-196 MR0839275
12 Shibata, Y.; Tsutsumi, Y. On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain Math. Z. 191 1986 165-199 MR0818663
13 Katayama, S. and Kubo, H., Lower bound of the lifespan of solutions to semilinear wave equations in an exterior domain, preprint (arXiv:1009.1188[math.AP])