Asymptotic Behavior of Solutions of Quasilinear Hyperbolic Equations with Linear Damping

We consider the asymptotic behavior of the solution of quasilinear hyperbolic equation with linear dampingV_tt-a(V_x)_x+αV _t=0,(x,t)∈R×(0,∞), ((*))subsequent to [K. Nishihara,J. Differential Equations131(1996), 171-188]. In that article, the system with dampingv_t-u_x=0,u_t+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 dampingV_tt+(p(v₀+V_x)-p(v₀)) _x+αV_t=0, and the decay rates V(t)_L ^∞=O(t^-1/2), V_x(t)_L ^∞=O(t^-1), V_t(t)_L ^∞=O(t^-3/2), were obtained provided that (V,V_t) _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), φ_x(t)_L ^∞=O(t^-1), φ_t(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.