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Weak asymptotic decay via a “relaxed invariance principle” for a wave equation with nonlinear, non-monotone damping

Published online by Cambridge University Press:  14 November 2011

M. Slemrod
M. Slemrod
Affiliation:
Center for Mathematical Sciences, University of Wisconsin, Madison, WI 53705, U.S.A.
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Synopsis

This paper considers the problem of asymptotic decay as t → ∞ of solutions of the wave equation utt − Δu = −a(x)β(ut, ∇u), (t,x) ∊ ℝ+ × Ω (a bounded, open, connected set in ℝN, N≧ 1, with smooth boundary), u =0 on ℝ+ × ∂Ω. The nonlinear function β is not assumed to be globally Lipschitz continuous, β(0, y2, …, yN+1) = 0, y1 β(y1…,yN+1) ≧ 0 for all y ∊ ℝN+1; β is not assumed to be monotone in y1. Under additional restrictions on the kernel of β, conditions are given which imply that [u, ut,] converges to [0,0] weakly in H = H10(Ω) × L2(Ω) as t → ∞. The work generalises earlier results of Dafermos and Haraux where strong decay in H as t → ∞ was obtained in the case β(y1 …, yN+1) = q(y1), q monotone on ℝ.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 1-2 , 1989 , pp. 87 - 97
Copyright
Copyright © Royal Society of Edinburgh 1989

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