On uniqueness and existence of entropy solutions for a nonlinear parabolic problem with absorption

Abstract

The aim of this paper is to study, for L 1-data, the absorption problem of parabolic type : $$u_t - {\rm div} a(u, Du) + \beta(x, u) \ni f$$ with Dirichlet boundary conditions and initial conditions. Here a satisfies the classical Leray-Lions hypotheses and β(x, ·) is the subdifferential ∂j(x, ·), where j is a convex function such that j(·, 0) = 0. Existence and uniqueness of an entropy solution is established.