Sharp viscous shock waves for relaxation model with degeneracy

This article is concerned with a relaxation model with degeneracy. Different from the existing studies on the regular viscous shock waves, we recognize that when the sub-characteristic condition is degenerate, the dynamical system possesses sharp viscous shock waves, a new type of viscous shock waves with sharp corners and being regionally degenerate in a half space. The regularities of these sharp viscous shock waves are further derived. Some of them can be smooth due to different degree of degeneracy, but most of them are Hölder continuous with sharp corners. Finally, the exact sharp viscous shock waves are constructed for three typical equations, and the corresponding numerical simulations with given initial data are also presented, which perfectly match the exact sharp viscous shock waves and also numerically demonstrate the stability of sharp shock waves.