caa a22 4500 386382840 CHVBK 20180307112048.0 cr unu---uuuuu 161130s1989 xx s 000 0 eng 10.1017/S0308210500023970 doi S0308210500023970 pii (NATIONALLICENCE)cambridge-10.1017/S0308210500023970 Slemrod M. Center for Mathematical Sciences, University of Wisconsin, Madison, WI 53705, U.S.A. Weak asymptotic decay via a "relaxed invariance principle” for a wave equation with nonlinear, non-monotone damping [Elektronische Daten] [M. Slemrod] 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. Copyright © Royal Society of Edinburgh 1989 Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 113/1-2(1989), 87-97 0308-2105 113:1-2<87 1989 113 PRM https://doi.org/10.1017/S0308210500023970 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0308210500023970 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Slemrod M. Center for Mathematical Sciences, University of Wisconsin, Madison, WI 53705, U.S.A NATIONALLICENCE 773 0- Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 113/1-2(1989), 87-97 0308-2105 113:1-2<87 1989 113 PRM CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge