L p -decay rates to nonlinear diffusion waves for p-system with nonlinear damping

Abstract

In this paper, we study the L p (2 ⩽ p ⩽ +∞) convergence rates of the solutions to the Cauchy problem of the so-called p-system with nonlinear damping. Precisely, we show that the corresponding Cauchy problem admits a unique global solution (v(x,t), u(x,t)) and such a solution tends time-asymptotically to the corresponding nonlinear diffusion wave (ῡ(x,t), ū(x,t)) governed by the classical Darcys’s law provided that the corresponding prescribed initial error function (w 0(x), z 0(x)) lies in (H 3 × H 2) (ℝ) and |v + − v −| + ∥w 0∥3 + ∥z 0∥2 is sufficiently small. Furthermore, the L p (2 ⩽ p ⩽ +∞) convergence rates of the solutions are also obtained.