Year: 2008
Journal of Partial Differential Equations, Vol. 21 (2008), Iss. 2 : pp. 141–172
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 ∈ 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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2008-JPDE-5275
Journal of Partial Differential Equations, Vol. 21 (2008), Iss. 2 : pp. 141–172
Published online: 2008-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 32
Keywords: Damped wave equation