DECAY RATES TOWARD STATIONARY WAVES OF SOLUTIONS FOR DAMPED WAVE EQUATIONS

摘要

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R+(utt-uxx+ut+f(u)x=0,t>0,z∈R+,(u(0,x)=u0(x)→u+, as x→+∞, (I)(ut(0,x)=u1(x),u(t,0)=ub.For the non-degenerate case,f'(u+)＜0,it is shown in[1]that the above initialboundary value problem admits a unique global solution u(t,x)which converges to the stationary wave φ(x)uniformly in x∈R+as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small.Moreover,by using the space-time weighted energy method initiated by Kawashima and Matsumura[2],the convergence rates(including the algebraic convergence rate and the exponential convergence rate)of u(t,x)toward φ(x)are also obtained in[1].We note,however,that the analysis in[1]relies heavily on the assumption that f'(ub)＜0.The main purpose of this paper is devoted to discussing the case of f'(ub)=0 and we show that similar results still hold for such a case.Our analysis is based on some delicate energy estimates.