Nonlinear Stability of Travelling Waves for a Hyperbolic System with Relaxation

We investigate the existence and the asymptotic stability of travelling wave solutions for a hyperbolic 2 2 system with a relaxation source term. Using the subcharacteristic stability condition, which implies special monotonicity properties of the solutions, we are able to establish the L 1 asymptotic attractivity of these solutions. In this paper we investigate nonlinear stability of travelling wave solutions to the following (semilinear) hyperbolic 2 2 system with a relaxation source term: (1.1) 8 > < > : @u @t + @v @x = 0 @v @t + @u @x = 1 " (f(u) ? v); (" > 0) where (x; t) 2 IR IR +. The unknowns u; v belong to IR, the function f = f(u) is in C 1 (IR) and > 0 is a constant to be xed later. Relaxation systems often arise in many physical situations. Let us recall, for example, gases not in thermodynamic equilibrium VK], kinetic theory Ce], CIP], PI], chromatography RAA], river ows, traac ows and more general waves Wh]. More recently the study of a special class of hyperbolic systems with relaxation was developed in view of the numerical approximation of discontinuous solutions of conservation laws JX], AN]. The 22 relaxation hyperbolic systems of conservation laws were rst analyzed by Liu Li2], who justiied some nonlinear stability criteria for diiusion waves, expansion waves and travelling waves.