Asymptotics for the nonlinear dissipative wave equation

Author:
Tokio Matsuyama

Journal:
Trans. Amer. Math. Soc. **355** (2003), 865-899

MSC (2000):
Primary 35L05; Secondary 35L10

DOI:
https://doi.org/10.1090/S0002-9947-02-03147-1

Published electronically:
November 1, 2002

MathSciNet review:
1938737

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We are interested in the asymptotic behaviour of global classical solutions to the initial-boundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a star-shaped obstacle. We shall treat the nonlinear dissipative term like , , and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as the time goes to infinity.

**1.**D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, 1983. MR**86c:35035****2.**N. Hayashi,*Global existence of small solutions to quadratic nonlinear wave equations in an exterior domain*, J. Funct. Anal.**131**(1995), 302-344. MR**96f:35113****3.**M. Ikawa, Hyperbolic partial differential equations and wave phenomena, Transl. Math. Monogr., Vol. 189, Amer. Math. Soc., 2000. MR**2001j:35176****4.**O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Revised 2nd ed., New York: Gordon and Breach, 1969. MR**40:7610****5.**J. L. Lions and W. A. Strauss,*Some nonlinear evolution equations*, Bull. Soc. Math. France**93**(1965), 43-96. MR**33:7663****6.**T. Matsuyama,*Asymptotic behaviour of solutions to the initial-boundary value problem with an effective dissipation around the boundary*, J. Math. Anal. Appl.**271**(2002), 467-492.**7.**T. Matsuyama,*Asymptotic behaviour of solutions for the nonlinear dissipative wave equations*, preprint (2001).**8.**S. Mizohata, The theory of partial differential equations, Cambridge Univ. Press, 1973. MR**58:29033****9.**K. Mochizuki,*Decay and asymptotics for wave equations with dissipative term*, Lecture Notes in Phys.**39**, 1975, Springer-Verlag, pp. 486-490. MR**58:29089****10.**K. Mochizuki, Scattering theory for wave equations (in Japanese), Kinokuniya, 1984.**11.**K. Mochizuki and T. Motai,*On energy decay-nondecay problems for the wave equations with nonlinear dissipative term in*, J. Math. Soc. Japan**47**(1995), 405-421. MR**96c:35122****12.**K. Mochizuki and H. Nakazawa,*Energy decay and asymptotic behavior of solutions to the wave equations with linear dissipation*, Publ. RIMS, Kyoto Univ.**32**(1996), 401-414. MR**97g:35101****13.**C. Morawetz,*Exponential decay of solutions of the wave equations*, Comm. Pure Appl. Math.**19**(1966), 439-444. MR**34:4664****14.**T. Motai and K. Mochizuki,*On asymptotic behaviors for wave equations with a nonlinear dissipative term in*, Hokkaido Math. J.**25**(1996), 119-135. MR**96m:35226****15.**M. Nakao,*Existence of global classical solutions of the initial-boundary value problem for some nonlinear wave equations*, J. Math. Anal. Appl.**146**(1990), 217-240. MR**91d:35141****16.**M. Nakao,*Stabilization of local energy in an exterior domain for the wave equation with a localized dissipation*, J. Differential Equations**148**(1998), 388-406. MR**2000c:35141****17.**J. Sather,*The existence of a global classical solution of the initial-boundary value problem for*, Arch. Rational Mech. Anal.**22**(1966), 129-135. MR**33:6124****18.**J. Shatah,*Global existence of small solutions to nonlinear evolution equations*, J. Differential Equations**46**(1982), 409-425. MR**84g:35036****19.**Y. Shibata and Y. Tsutsumi,*Global existence theorem of nonlinear wave equations in the exterior domain*, Lecture Notes in Num. Appl. Anal.**6**(1983), 155-196, Kinokuniya/North-Holland. MR**87f:35161****20.**Y. Shibata and Y. Tsutsumi,*On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain*, Math. Z.**191**(1986), 165-199. MR**87i:35122**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35L05,
35L10

Retrieve articles in all journals with MSC (2000): 35L05, 35L10

Additional Information

**Tokio Matsuyama**

Affiliation:
Department of Mathematics, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan

Email:
matsu@sm.u-tokai.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-02-03147-1

Received by editor(s):
November 6, 2001

Received by editor(s) in revised form:
July 4, 2002

Published electronically:
November 1, 2002

Additional Notes:
Supported in part by a Grant-in-Aid for Scientific Research (C)(2)(No.11640213), Japan Society for the Promotion of Science.

The author would like to express his sincere gratitude to Professors K. Mochizuki, M. Nakao and M. Yamaguchi for several useful comments. He is also indebted to Professors Y. Shibata, N. Hayashi and T. Kobayashi, who pointed out the uniform decay estimate to him. The author thanks Doctor H. Nakazawa for advising him of the existence of scattering states. The author also thanks the referee for a careful reading of the manuscript.

Dedicated:
Dedicated to Professor Kunihiko Kajitani on the occasion of his sixtieth birthday

Article copyright:
© Copyright 2002
American Mathematical Society