TY - JOUR
T1 - Decay Rates Toward Stationary Waves of Solutions for Damped Wave Equations
JO - Journal of Partial Differential Equations
VL - 2
SP - 141
EP - 172
PY - 2008
DA - 2008/05
SN - 21
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5275.html
KW - Damped wave equation
KW - stationary wave
KW - asymptotic stability
KW - decay rates
KW - space-time weighted energy method
AB -  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 ∈ R_+, u(0,x)=u_0(x)→ u_+, as x→+∞, 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 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'(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.