CONVERGENCE RATES TO ASYMPTOTIC PROFILE FOR SOLUTIONS OF QUASILINEAR HYPERBOLIC EQUATIONS WITH LINEAR DAMPING

Abstract

This paper is concerned with the asymptotic behavior of the solution of quasilinear hyperbolic equations with linear damping. The main novelty lies in the following observation: If we suitably choose the initial data of the corresponding parabolic equation, then the solution Ψ = Ψ(x, t) of the parabolic equation served as the new asymptotic profile satisfies ‖(V-Ψ, (V-Ψ)x, (V-Ψ)t)(t)‖L∞ = O(1)(t-2, t-5/2, t-3). The convergence rates of the new profile Ψ are better than that obtained by H.-J. Zhao (2000, J. Differential Equations167, 467–494), and we need none of the additional technical assumptions (H1) and (H2) therein. Therefore, we answer an open problem posed by Nishihara (1997, J. Differential Equations133, 384–395).