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Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent

Published online by Cambridge University Press:  14 November 2011

Eduard Feireisl
Eduard Feireisl
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain
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Extract

We prove the existence of a global attractor for the problem

where f is ‘coercive at infinity’ and satisfies the growth condition

while g ϵL2(R3).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 5 , 1995 , pp. 1051 - 1062
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Babin, A. V. and Vishik, M. I.. Attractors of partial differential evolution equations in an unbounded domain. Proc. Roy. Soc. Edinburgh Sect. A 116 (1990), 221–43.Google Scholar
2Babin, A. V. and Vishik, M. I.. Attractors of Evolution Equations (Amsterdam: North Holland, 1992).Google Scholar
3Brenner, P.. On space-time means and strong global solutions of nonlinear hyperbolic equations. Math. Z. 201 (1989), 45–55.CrossRefGoogle Scholar
4Feireisl, E.. Attractors for semilinear damped wave equations on R3. Nonlinear Anal. 23 (1994), 187–95.Google Scholar
5Ghidaglia, J. M. and Temam, R.. Attractors for damped nonlinear hyperbolic equations. J. Math. Pures Appl. 66 (1987), 273–319.Google Scholar
6Ginibre, J. and Velo, G.. The global Cauchy problem for the nonlinear Klein-Gordon equation. Math. Z. 189 (1985), 487–505.CrossRefGoogle Scholar
7Ginibre, J., Soffer, A. and Velo, G.. The global Cauchy problem for the critical nonlinear wave equation. J. Funct. Anal. 110 (1992), 96–130.Google Scholar
8Grillakis, M.. Regularity for the wave equation with critical nonlinearity. Comm. Pure Appl. Math. 45(1992), 749–74.Google Scholar
9Hale, J. K.. Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25 (Providence, RI: American Mathematical Society, 1988).Google Scholar
10Haraux, A.. Systèmes Dynamiques Dissipatifs et Applications, RMA 17 (Paris: Masson, 1991).Google Scholar
11Jorgens, K.. Das Anfangswertproblem im Grossen fur eine Klasse nichtlinearer Wellengleichungen. MatkZ. 77 (1961), 295–308.Google Scholar
12Kapitanskii, L. V.. Cauchy problem for a semilinear wave equation III. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 181 (1990), 24–64 [in Russian].Google Scholar
13Ruiz, A.. Unique continuation for weak solutions of the wave equation plus a potential (preprint, Universidad Autonoma Madrid, 1988).Google Scholar
14Shatah, J. and Struwe, M.. Regularity results for nonlinear wave equations (preprint, 1991).Google Scholar
15Strauss, W. A.. Nonlinear Wave Equations, Conference Board and Mathematical Sciences 73 (Providence, Rl: American Mathematical Society, 1989).Google Scholar
16Strichartz, R. S.. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977), 705–14.Google Scholar
17Struwe, M.. Semilinear wave equations. Bull. Amer. Math. Soc. 26(1) (1992), 53–85.CrossRefGoogle Scholar
18Temam, R.. Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences 68 (Berlin: Springer, 1988).Google Scholar
19Zuazua, E.. Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures Appl. 70 (1991), 513–29.Google Scholar