Weak entropy solutions for degenerate parabolic–hyperbolic inequalities

Abstract

We study inner obstacle problems for a class of strongly degenerate parabolic–hyperbolic quasilinear operators associated with Dirichlet data in an open bounded subset of ${\mathbb{R}}^p$, $p\ge 1$. We first give the definition of a weak entropy solution that warrants uniqueness; the boundary conditions are expressed by using the framework of divergence measure fields. The existence of such a solution is obtained through the vanishing viscosity method.