Large time behavior of solutions to a semilinear hyperbolic system with relatxaion

We are concerned with the initial value problem for a damped wave equation with a nonlinear convection term which is derived from a semilinear hyperbolic system with relaxation. We show the global existence and asymptotic decay of solutions in W^1,p (1 ≤ p ≤ ∞) under smallness condition on the initial data. Moreover, we show that the solution approaches in W^1,p (1 ≤ p ≤ ∞) the nonlinear diffusion wave expressed in terms of the self-similar solution of the Burgers equation as time tends to infinity. Our results are based on the detailed pointwise estimates for the fundamental solutions to the linearlized equation.