Time decay rate of solutions toward the viscous shock waves to the Cauchy problem for the scalar conservation law with nonlinear viscosity and discontinuous initial data

Abstract

In this paper, it is shown that the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear viscosity and different far field states, exists globally in time. Especially, this existence holds with discontinuous initial data. Furthermore, when the corresponding Riemann problem for the hyperbolic part has a single shock wave solution, it is proved that the weak solution converges toward the viscous shock wave as time goes to infinity, and the decay rate is also obtained. The proof is given by a technical energy method.