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

Published electronically:
November 1, 2002

MathSciNet review:
1938737

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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.**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190****2.**Nakao Hayashi,*Global existence of small solutions to quadratic nonlinear wave equations in an exterior domain*, J. Funct. Anal.**131**(1995), no. 2, 302–344. MR**1345034**, 10.1006/jfan.1995.1091**3.**Mitsuru Ikawa,*Hyperbolic partial differential equations and wave phenomena*, Translations of Mathematical Monographs, vol. 189, American Mathematical Society, Providence, RI, 2000. Translated from the 1997 Japanese original by Bohdan I. Kurpita; Iwanami Series in Modern Mathematics. MR**1756774****4.**O. A. Ladyzhenskaya,*The mathematical theory of viscous incompressible flow*, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. MR**0254401****5.**Jacques-Louis Lions and W. A. Strauss,*Some non-linear evolution equations*, Bull. Soc. Math. France**93**(1965), 43–96. MR**0199519****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.**Sigeru Mizohata,*The theory of partial differential equations*, Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara. MR**0599580****9.**Kiyoshi Mochizuki,*Decay and asymptotics for wave equations with dissipative term*, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975) Springer, Berlin, 1975, pp. 486–490. Lecture Notes in Phys., 39. MR**0600342****10.**K. Mochizuki, Scattering theory for wave equations (in Japanese), Kinokuniya, 1984.**11.**Kiyoshi Mochizuki and Takahiro Motai,*On energy decay-nondecay problems for wave equations with nonlinear dissipative term in 𝑅^{𝑁}*, J. Math. Soc. Japan**47**(1995), no. 3, 405–421. MR**1331322**, 10.2969/jmsj/04730405**12.**Kiyoshi Mochizuki and Hideo Nakazawa,*Energy decay and asymptotic behavior of solutions to the wave equations with linear dissipation*, Publ. Res. Inst. Math. Sci.**32**(1996), no. 3, 401–414. MR**1409795**, 10.2977/prims/1195162849**13.**Cathleen S. Morawetz,*Exponential decay of solutions of the wave equation*, Comm. Pure Appl. Math.**19**(1966), 439–444. MR**0204828****14.**Takahiro Motai and Kiyoshi Mochizuki,*On asymptotic behaviors for wave equations with a nonlinear dissipative term in 𝐑^{𝐍}*, Hokkaido Math. J.**25**(1996), no. 1, 119–135. MR**1376496**, 10.14492/hokmj/1351516713**15.**Mitsuhiro Nakao,*Existence of global classical solutions of the initial-boundary value problem for some nonlinear wave equations*, J. Math. Anal. Appl.**146**(1990), no. 1, 217–240. MR**1041212**, 10.1016/0022-247X(90)90343-E**16.**Mitsuhiro Nakao,*Stabilization of local energy in an exterior domain for the wave equation with a localized dissipation*, J. Differential Equations**148**(1998), no. 2, 388–406. MR**1643195**, 10.1006/jdeq.1998.3468**17.**Jerome Sather,*The existence of a global classical solution of the initial-boundary value problem for 𝑐𝑚𝑢+𝑢³=𝑓*, Arch. Rational Mech. Anal.**22**(1966), 292–307. MR**0197965****18.**Jalal Shatah,*Global existence of small solutions to nonlinear evolution equations*, J. Differential Equations**46**(1982), no. 3, 409–425. MR**681231**, 10.1016/0022-0396(82)90102-4**19.**Yoshihiro Shibata and Yoshio Tsutsumi,*Global existence theorem for nonlinear wave equation in exterior domain*, Recent topics in nonlinear PDE (Hiroshima, 1983) North-Holland Math. Stud., vol. 98, North-Holland, Amsterdam, 1984, pp. 155–196. MR**839275**, 10.1016/S0304-0208(08)71498-1**20.**Yoshihiro Shibata and Yoshio Tsutsumi,*On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain*, Math. Z.**191**(1986), no. 2, 165–199. MR**818663**, 10.1007/BF01164023

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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