Homogeneous Dirichlet Problem For General Anisotropic Degenerate Parabolic-Hyperbolic Equations

The aim of this paper is to prove the well-posedness(existence and uniqueness) of entropy solutions to the initial-boundary problem of the general anisotropic de-generate parabolic-hyperbolic equations with (t, x) dependent coefficients for L∞ini-tial data and homogeneous Dirichlet boundary condition. The research on degenerate parabolic-hyperbolic equations mainly focus on the isotropic diffusion case. For the anisotropic diffusion case, Chen-Perthame[1]in 2003 first proved the existence and uniqueness of L1 kinetic solution for the Cauchy problem by introducing a kinetic formulation. In 2005, Chen-Karlsen[2]proved the existence and uniqueness of the L∞entropy solution for the Cauchy problem in the case that the coefficients depend on (t,x). While for the initial-boundary value problem, Li-Wang[3] first established the existence and uniqueness of L∞entropy solution for the homogeneous Dirichlet problem with (t,x) independent coefficients. Based on these, we study the general case with (t,x) dependent coefficients and establish the existence and uniqueness of L∞entropy solution. We introduce the entropy-entropy flux triples and the boundary entropy-entropy flux triples, define the entropy solutions, and use the Kruzkov's device of doubling variables to show the uniqueness result and prove the existence result by using the vanishing viscosity method. We mainly focus on the proof of the uniqueness result, while the arguments for the boundary is the key and difficult point in this paper.