Abstract :
We consider the asymptotic behavior of the solution of quasilinear hyperbolic equation with linear dampingVtt−a(Vx)x+αVt=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)(±∞)=( 0, 0) and ∫∞−∞ (v0(x)− 0) dx=0 this system is reduced to the quasilinear hyperbolic equation with linear dampingVtt+(p( 0+Vx)−p( 0))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.
