Asymptotic Behavior of Solutions to a Hyperbolic System with Relaxation and Boundary Effect

Abstract

In this paper, we study the initial boundary value problem of the following hyperbolic system with relaxation {vt + Ux = 0, ut + σ(v)x = 1/ε (f(v) - u), (x, t) ∈ R+× R+ (E) on the half line R+ with the boundary conditions v(0, t) = v_. When the asymptotic states are stationary wave or rarefaction wave or superposition of these two kind waves, we prove the stability of these wave patterns for small perturbation. The study is motivated by [7] where the asymptotic behavior of solutions to the scalar viscous conservation law with boundary corresponding to rarefaction waves was studied. In our analysis, we do not require (v+, u+) satisfy the equilibrium equation, i.e., u+= f(v+) as in [15]. © 2000 Academic Press.