Nonlinear stability of viscous shock profiles for a hyperbolic system with Cattaneo's law in one‐dimensional half space

Abstract

We consider the asymptotic nonlinear stability of viscous shock profiles for an initial‐boundary value problem of the scalar conservation laws with an artificial heat flux satisfying Cattaneo's law in the negative half line with Dirichlet boundary condition. When the nonlinear flux function is assumed to be strictly convex and the unique global entropy solution of the corresponding Riemann problem of the resulting scalar conservation laws consists of shock wave with negative speed, it is shown in this paper that the large time behavior of its global smooth solutions can be precisely described by the suitably shifted viscous shock profiles, where the time‐dependent shift function is uniquely determined by both the boundary value and the initial data. We also show that the shift function converge to a constant time asymptotically. Our analysis is based on weighted energy method.