Asymptotics for the nonlinear dissipative wave equation
Author:
Tokio Matsuyama
Journal:
Trans. Amer. Math. Soc. 355 (2003), 865899
MSC (2000):
Primary 35L05; Secondary 35L10
Published electronically:
November 1, 2002
MathSciNet review:
1938737
Fulltext PDF Free Access
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Abstract: We are interested in the asymptotic behaviour of global classical solutions to the initialboundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a starshaped 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., SpringerVerlag, 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), 302344. 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
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O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Revised 2nd ed., New York: Gordon and Breach, 1969. MR 40:7610
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J. L. Lions and W. A. Strauss, Some nonlinear evolution equations, Bull. Soc. Math. France 93 (1965), 4396. MR 33:7663
 6.
T. Matsuyama, Asymptotic behaviour of solutions to the initialboundary value problem with an effective dissipation around the boundary, J. Math. Anal. Appl. 271 (2002), 467492.
 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, SpringerVerlag, pp. 486490. MR 58:29089
 10.
K. Mochizuki, Scattering theory for wave equations (in Japanese), Kinokuniya, 1984.
 11.
K. Mochizuki and T. Motai, On energy decaynondecay problems for the wave equations with nonlinear dissipative term in , J. Math. Soc. Japan 47 (1995), 405421. 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), 401414. MR 97g:35101
 13.
C. Morawetz, Exponential decay of solutions of the wave equations, Comm. Pure Appl. Math. 19 (1966), 439444. 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), 119135. MR 96m:35226
 15.
M. Nakao, Existence of global classical solutions of the initialboundary value problem for some nonlinear wave equations, J. Math. Anal. Appl. 146 (1990), 217240. 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), 388406. MR 2000c:35141
 17.
J. Sather, The existence of a global classical solution of the initialboundary value problem for , Arch. Rational Mech. Anal. 22 (1966), 129135. MR 33:6124
 18.
J. Shatah, Global existence of small solutions to nonlinear evolution equations, J. Differential Equations 46 (1982), 409425. 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), 155196, Kinokuniya/NorthHolland. 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), 165199. MR 87i:35122
 1.
 D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., SpringerVerlag, 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), 302344. 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), 4396. MR 33:7663
 6.
 T. Matsuyama, Asymptotic behaviour of solutions to the initialboundary value problem with an effective dissipation around the boundary, J. Math. Anal. Appl. 271 (2002), 467492.
 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, SpringerVerlag, pp. 486490. MR 58:29089
 10.
 K. Mochizuki, Scattering theory for wave equations (in Japanese), Kinokuniya, 1984.
 11.
 K. Mochizuki and T. Motai, On energy decaynondecay problems for the wave equations with nonlinear dissipative term in , J. Math. Soc. Japan 47 (1995), 405421. 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), 401414. MR 97g:35101
 13.
 C. Morawetz, Exponential decay of solutions of the wave equations, Comm. Pure Appl. Math. 19 (1966), 439444. 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), 119135. MR 96m:35226
 15.
 M. Nakao, Existence of global classical solutions of the initialboundary value problem for some nonlinear wave equations, J. Math. Anal. Appl. 146 (1990), 217240. 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), 388406. MR 2000c:35141
 17.
 J. Sather, The existence of a global classical solution of the initialboundary value problem for , Arch. Rational Mech. Anal. 22 (1966), 129135. MR 33:6124
 18.
 J. Shatah, Global existence of small solutions to nonlinear evolution equations, J. Differential Equations 46 (1982), 409425. 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), 155196, Kinokuniya/NorthHolland. 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), 165199. MR 87i:35122
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Additional Information
Tokio Matsuyama
Affiliation:
Department of Mathematics, Tokai University, Hiratsuka, Kanagawa 2591292, Japan
Email:
matsu@sm.utokai.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994702031471
PII:
S 00029947(02)031471
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 GrantinAid 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
