Entropy Solutions for Nonlinear Degenerate Problems

Abstract

. We consider a class of elliptic‐hyperbolic degenerate equations $g(u)-\Delta b(u) +\divg\phi (u) =f$ with Dirichlet homogeneous boundary conditions and a class of elliptic‐parabolic‐hyperbolic degenerate equations $g(u)_t-\Delta b(u) +\divg\phi (u) =f$ with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function φ satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for φ continuous.