Asymptotic Behavior of Solutions of Quasilinear Hyperbolic Equations with Linear Damping

Abstract

We consider the asymptotic behavior of the solution of quasilinear hyperbolic equation with linear dampingVtt-a(Vx)x+αV t=0,(x,t)∈R×(0,∞), ((*))subsequent to [K. Nishihara,J. Differential Equations131(1996), 171-188]. In that article, the system with dampingvt-ux=0,ut+p(v) x=-αu,p′(v)<0(v>0) was treated, and the convergence rates to the diffusion wave by [L. Hsiao and T.-P. Liu,Comm. Math. Phys.143(1992), 599-605 and L. Hsiao and T.-P. Liu,Chinese Ann. Math. Ser. B14(1993), 465-480] were improved. For initial data (v0,u0)(x) satisfying (v0,u0)(±∞)=(v0,0) and ∫∞-∞(v0(x)-v0)dx=0 this system is reduced to the quasilinear hyperbolic equation with linear dampingVtt+(p(v0+Vx)-p(v0)) x+αVt=0, and the decay rates V(t)L∞=O(t-1/2), Vx(t)L∞=O(t-1), Vt(t)L∞=O(t-3/2), were obtained provided that (V,Vt) t=0∈(H3×H2) is small and inL1×L1. From these decay rates,Vis expected to behave as a solution to the parabolic equation. The aims of this paper are to seek the asymptotic profileφ(x,t) satisfying φ(t)L∞=O(t-1/2), φx(t)L∞=O(t-1), φt(t)L∞=O(t-3/2), and to show that the solution of (*) satisfies (V-φ)(t)L∞=O(t-1), (V-φ)x(t)L∞=O(t-3/2), (V-φ)t(t)L∞=O(t-2). The proof is based on the elementary energy method and the Green function method for the parabolic equation. © 1997 Academic Press.