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

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 (nu (0)(x), u(0)(x)), the Cauchy problem (1.1), (1.2) admits a unique global smooth solution (nu (t, x), u(t, x)) and such a solution tends time-asymptotically, at the optimal L-P(2 less than or equal to p less than or equal to infinity) decay rates, to the corresponding nonlinear diffusion wave ( )over bar>)(t, x), (u) over bar (t, x)) governed by the classical Darcy's law provided that the corresponding prescribed initial error function (V-0(x), U-0(x)) ties in (H-3 x H-2) (R) boolean AND (L-1 x L-1)(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 Equations 131, 171-188), and K. Nishihara, W.-K. Wang, and T. Yang (2000, J. Differential Equations 161, 191-218), our main novelty lies in the facts that the nonlinear diffusion wave ( )over bar>(t, x), (u) over bar (t, x)) need not to be weak and (V-0(x), U-0(x)) can be chosen arbitrarily large. Secondly, we show that the nonlinear diffusion waves ( )over bar>(t, x), (u) over bar (t, x)) are nonlinear stable provided that the strength or the nonlinear diffusion waves is weak and that the initial disturbance (V-0(x), U-0(x)) satisfies the assumption that parallel toV(0xx)(x)parallel to (L infinity) + parallel toU(0x)(x)parallel to (L infinity) is Sufficiently small. We also show that the smallness assumption imposed oil 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. (C) 2001 Academic Press.