The Renormalized Entropy Solutions Of Homogeneous Dirichlet Problem For Anisotropic Degenerate Parabolic-Hyperbolic Equations

In this paper, we study the well-posedness of renormalized entropy solution of ho-mogeneous Dirichlet problem for anisotropic degenerate parabolic-hyperbolic equa-tions with the coefficients depending on (t, x). This type of equations is appliedin many fields, such as the migration of pollution in porous medias, heat conduc-tion, financial decision-making process and so on. We introduce the entropy-entropyflux triples and the boundary entropy-entropy flux triples and present the definitionof entropy solution.When the initial data belongs to L1space, the entropy solutionof this homogeneous Dirichlet problem may be unbounded. Meanwhile, the localLipschitz continuity of the convection and diffusion functions may lead to their lo-cal non-integrability. We introduce the renormalized entropy solution, and then provethe existence and uniqueness by vanishing viscosity method and Kruzˇkov’s device ofdoubling variables, respectively.