Asymptotic Behaviors of Solutions of Quasilinear Hyperbolic Equations with Linear Damping II

Abstract

This paper, as a follow-up to the recent work of K. Nishihara (1997, J. Differential Equations133, 384–395), is concerned with the asymptotic behaviors of the solution of quasilinear hyperbolic equation with linear damping Vtt−a(Vx)x+Vt=0,a′(v)>0forallv∈R,which satisfies the following prescribed initial condition:(V(t, x), Vt(t, x))|t=0=(V0(x), V1(x)),x∈R.Compared with the results obtained by K. Nishihara, the main novelties of our present paper lie in the following: First, we recover all the results obtained by K. Nishihara under some other smallness assumptions on the initial data (V0(x), V1(x)) which are much weaker than those needed in K. Nishihara's arguments. Second, under some additional assumptions on the nonlinear function a(v) and on the initial data (V0(x), V1(x)), when δ0=0, δ1≠0, we succeed in proving that φ(t, x) is still an asymptotic profile of V(t, x) and this answers partially the open problem left by K. Nishihara.