References
Chadam, J.: Asymptotic for□u=m 2u+G(t, x, u, ux, ut). I. Ann. Scuola Norm. Sup., Pisa26, 33–65 (1972)
Dionne, P.: Sur les problème de Cauchy hyperboliques bién poses. J. Analyse Math.10, 1–90 (1962)
Ikawa, M.: Mixed problems for hyperbolic equations of second order. J. Math. Soc. Japan20, 580–608 (1968)
John, F.: Blow-up for quasi-linear wave equations in three space dimension. Commun. Pure Appl. Math.34, 29–51 (1981)
Klainerman, S.: Global existence for nonlinear wave equations. ibid.33, 43–101 (1980)
Klainermann, S., Ponce, G.: Global small amplitude solutions to nonlinear evolution equations. Ibid.36, 133–141 (1983)
Lax, P. D., Phillips, R. S.: Scattering Theory. New York & London: Academic Press 1967
Lax, P. D., Morawetz, C. S., Phillips, R. S.: Exponential decay of solutions of the wave equations in the exterior of a star-shaped obstacle. Ibid.. New York & London: Academic Press16, 477–486 (1963)
Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ.20, 67–104 (1980)
Melrose, R. B.: Singularities and energy decay in acoustical scattering. Duke Math. J.46, 43–59 (1979)
Mizohata, S.: The theory of partial differential equations. London: Cambridge Univ., Press 1973
Mizohata, S.: Quelque problemès au bord, du type mixte, pour des équations hyperboliques, Séminair sur les équations aux derivées partielles. Collège de France, 23–60 (1966/67)
Rabinowitz, P. H.: Periodic solutions of nonlinear hyperbolic partial differential equations II. Commun. Pure Appl. Math.22, 15–39 (1969)
Segal, I.: Non-linear semi-groups. Ann. Math.78, 339–364 (1963)
Shatah, J.: Global existence of small solutions to nonlinear evolution equations. J. Diff. Eqs.46, 409–425 (1982)
Shibata, Y.: On the global existence theorem of classical solutions of mixed problem for some second order non-linear hyperbolic operators with dissipative term in the interior domain. Funk. Ekva.25, 303–345 (1982)
Shibata, Y.: On the global existence theorem of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain. Tsukuba J. Math.7, 1–68 (1983)
Shibata, Y., Tsutsumi, Y.: Global existence theorem of nonlinear wave equation in the exterior domain. Lecture Notes in Num. Appl. Anal.6, 155–196 (1983), Kinokuniya/North-Holland
Shibata, Y., Tsutsumi, Y.: Local existence of solution for the initial-boundary value problem of fully nonlinear wave equation. To appear
Tsutsumi, Y.: Global solutions of the nonlinear Schrödinger equation in exterior domains. Commun. P.D.E.8, 1337–1374 (1983)
Tsutsumi, Y.: Local energy decay of solutions to the free Schrödinger equation in exterior domain. J. Fac. Sci. Univ. Tokyo Sec. IA,31, 97–108 (1984)
Vainberg, B. R.: On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour ast→∞ of solutions of nonstationary problems. Russian Math. Survey30 (2), 1–58 (1975)
von Wahl, W.:L p-decay rates for homogeneous wave equations. Math. Z.120, 93–106 (1971)
Yamamoto, K.: Characterization of convex obstacle by singularities of the scattering kernel. To appear
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shibata, Y., Tsutsumi, Y. On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain. Math Z 191, 165–199 (1986). https://doi.org/10.1007/BF01164023
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01164023