Nonhomogeneous Dirichlet Problems for Degenerate Parabolic-Hyperbolic Equations

Abstract

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 a ₀. Here u=u(x,t)∈ℝ, with (x,t)∈Q=Ω× (0,T), where Ω is a bounded domain in ℝ^N with smooth boundary and T>0. The function b is assumed to be nondecreasing (allowing degeneration zones where b is constant), Φ is locally Lipschitz continuous and g∈L ^∞(Ω× (0,b)). After introducing the definition of an entropy solution to the above problem (in the spirit of Otto [14]), 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 (specifically the diffusion function b is approximated by functions b ^ɛ that are strictly increasing).

The approach proposed for the hyperbolic-parabolic problem can be used to prove similar results for the class of hyperbolic-elliptic boundary value problems of the form

again in the case of nonconstant boundary data a ₀.