Convergence to Strong Nonlinear Diffusion Waves for Solutions of p-System with Damping

Abstract

This paper is concerned with the p-system with frictional damping and our main purpose is two-fold: First, we show that for a certain class of given large initial data (v0(x), u0(x)), the Cauchy problem (1.1), (1.2) admits a unique global smooth solution (v(t, x), u(t, x)) and such a solution tends time-asymptotically, at the optimal Lp(2⩽p⩽∞) decay rates, to the corresponding nonlinear diffusion wave (v(t, x), u(t, x)) governed by the classical Darcy's law provided that the corresponding prescribed initial error function (V0(x), U0(x)) lies in (H3×H2)(R)∩(L1×L1)(R). Compared with former results in this direction obtained by L. Hsiao and T.-P. Liu (1992, Comm. Math. Phys.143, 599–605), K. Nishihara (1996, J. Differential Equations131, 171–188), and K. Nishihara, W.-K. Wang, and T. Yang (2000, J. Differential Equations161, 191–218), our main novelty lies in the facts that the nonlinear diffusion wave (v(t, x), u(t, x)) need not to be weak and (V0(x), U0(x)) can be chosen arbitrarily large. Secondly, we show that the nonlinear diffusion waves (v(t, x), u(t, x)) are nonlinear stable provided that the strength of the nonlinear diffusion waves is weak and that the initial disturbance (V0(x), U0(x)) satisfies the assumption that ‖V0xx(x)‖L∞+‖U0x(x)‖L∞ is sufficiently small. We also show that the smallness assumption imposed on the strength of the diffusion waves is a necessary condition to guarantee the nonlinear stability result and compared with the corresponding results obtained by L. Hsiao and T.-P. Liu (1992, Comm. Math. Phys.143, 599–605), the smallness conditions we imposed on the initial disturbance are much more weaker.