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

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.