Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations
Résumé
The aim of the paper is to give a formulation for the initial boundary value problem of parabolic-hyperbolic type in the case of nonhomogeneous boundary data. The function diffusion function is assumed to be nondecreasing (allowing degeneration zones where it is constant) and is locally Lipschitz continuous. After introducing the definition of an entropy solution to the above problem, we prove uniqueness of the solution in the proposed setting. Moreover we prove that the entropy solution previously defined can be obtained as the limit of solutions of regularized equations of nondegenerate parabolic type. The approach proposed for the hyperbolic-parabolic problem can be used to prove similar results for the class of hyperbolic-elliptic boundary value problems, again in the case of nonconstant boundary data a0.