CONVERGENCE RATES TO NONLINEAR DIFFUSION WAVES FOR p-SYSTEM WITH NONLINEAR DAMPING ON QUADRANT

In this paper, we study the asymptotic behavior and the convergence rates of solutions to the so-called p-system with nonlinear damping on quadrant R+ x R+ = (0, infinity) x (0, infinity), v(t) - u(x) = 0, u(t) + p(v)(x) = -alpha u - g(u) with the Dirichlet boundary condition u|(x=0) = 0 or the Neumann boundary condition u(x)|(x=0) = 0. The initial data (v(0), u(0))(x) has the constant sales (v(+), u(+)) at x = infinity. In the case of null-Dirichlet boundary condition on u, we show that the corresponding 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 ((v) over bar (x,t), (u) over bar (x,t)) governed by the classical Darcy's law provided that the corresponding prescribed initial error function (w(0)(x), z(0)(x)) lies in (H-3 x H-2) (R+) and ||v(0)(x) - v+||(L)1 + ||w(0)||(3) + ||z(0)||(2) + ||V-0||(5) + ||Z(0)||(4) is sufficiently small. Its optimal L-infinity convergence rate is also obtained by using the Green function of the diffusion equation. In the case of null-Neumann boundary condition on u, the global existence of smooth solution with small initial data is obtained in both of the case of v(0)(0) = v(+) and v(0)(0) not equal v(+). The solution (v(x,t), u(x,t)) is proved to tend to ((v) over bar (x,t),0) as t tends to infinity, and we also get the optimal L-infinity convergence rate in the case of v(0)(0) = v(+).