Weak Asymptotic Decay via a 'Relaxed Invariance Principle' for a Wave Equation with Nonlinear, Nonmonotone Damping

摘要

This paper considers the problem of asymptotic decay as t approaches infinity of solutions of the wave equation u sub tt - delta u = -a(x) beta(u sub t,gradu), (t,x) is an element of R+X Omega (a bounded, open, connected set in R to the Nth power, N > or = 1, with smooth boundary), u = O on R+ X del Omega. The nonlinear function beta is assumed to be globally Lipschitz continuous, beta(y) = o(abs. val. y) as abs. val. y approaches infinity, beta(O,y2,...,y(N+1) = O, y1 beta (y1,...,y(N+1) > or = for all y an element of R to the N + 1 power; beta is not assumed to be monotone in y1. Under additional restrictions on the kernel of beta conditions are given which imply U,UT converges to 0,0 weakly in H = H sub 0 superscript 1 (Omega) x L superscript 2 (Omega) as t approaches infinity. The work generalizes earlier results of where strong decay in H as t approaches infinity was obtained in the case beta(y1,...,y(N+1) = q(y1), q monotone R. Keywords: Wave equation, Invariance principle, Young measure. (jhd)