Long‐time behaviour of diffusion waves for a viscous system of conservation laws in $\mathbf{R}^N$

Abstract

We consider a system of two viscous conservation laws in several space dimensions. The equations are weakly coupled so that the inviscid part of the system is not necessarily strictly hyperbolic. We analyze the long‐time behaviour of integrable non‐negative solutions. More precisely, we are interested by the profile of the first non‐trivial term of the asymptotic expansion of the solutions. The flux function may fail to be $C^2$ , thus linearization around the zero solution does not apply. We use scaling techniques. Inspired in our previous works on scalar equations we exhibit two different types of behaviour: weakly nonlinear and self‐similar.