Decay Rates Toward Stationary Waves of Solutions for Damped Wave Equations

Abstract

" This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R_+ u_{tt}-u_{xx}+u_t+f(u)_x=0, t > 0, x \u2208 R_+, u(0,x)=u_0(x)\u2192 u_+, as x\u2192+\u221e, u_t(0,x)=u_1(x), u(t,0)=u_b. 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 \u03c6\u001e(x) uniformly in x \u2208 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 \u001e\u03c6(x) are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f'(u_b) < 0. The main purpose of this paper is devoted to discussing the case of f'(u_b) = 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates."