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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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