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Compact attractor for a nonlinear wave equation with critical exponent

Published online by Cambridge University Press:  14 November 2011

Orlando Lopes
Orlando Lopes
Affiliation:
Departamento de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, 13081 – Campinas – SP, Brasil
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Synopsis

In this paper we study the existence of a compact attractor for the solutions of the equation utt − Δu + cut + f(u) = h(t, x), x ∊ ℝ3. The phase space is H1 × L2 and periodicity in the x-variables is taken as a boundary condition. Besides the usual coercive condition, we assume f satisfies the growth condition |f′(u)|≦ a + bu2; this growth condition is critical because the embedding H1 → L6 is not compact. In the proof we use an Lp − H1.q estimate for the linear homogeneous wave equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 1-2 , 1990 , pp. 61 - 64
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Babin, A. V. and Vishik, M. I.. Regular attractors of semigroups and evolution equations. J. Math. Pures Appl. 62 (1983), 441–491.Google Scholar
2Haraux, A.. Two remarks on dissipative hyperbolic problems. In Seminaire du College de France, ed. Lions, J. L. (Boston: Pitman, 1985).Google Scholar
3Ghidaglia, J. M. and Temam, R.. Attractors for damped nonlinear hyperbolic equations. J. Math. Pures Appl. 66 (1987), 273–320.Google Scholar
4Hale, J.. Asymptotic behavior of dissipative systems (Providence, R.I.: American Mathematical Society, 1988).Google Scholar
5Lopes, O.. Asymptotic behavior of solutions of a nonlinear wave equation. Bol. Soc. Bras. Mat. 19 (1988), 141–150.CrossRefGoogle Scholar
6Webb, G. F.. A bifurcation problem for a nonlinear hyperbolic PDE. SIAM J. Math. Anal. 10 (1979), 922–932.CrossRefGoogle Scholar
7Ball, J.. On the asymptotic behavior of generalized process with applications to nonlinear evolution equations. J. Differential Equations 27 (1978), 224–265.CrossRefGoogle Scholar