Well-posedness of singular diffusion equations in porous media with homogeneous Neumann boundary conditions

Abstract

The paper deals with the study of the well-posedness of diffusion equations with homogeneous Neumann boundary conditions. Three cases of singular diffusion equations, fast, slow and very fast, with nonlinear transport terms are considered. The fast and slow cases involve differential inclusions. The problem with homogeneous Neumann boundary conditions has been still open for the multivalued singular cases of diffusion analyzed here and especially when transport terms are present. Particular hypotheses imposed to the transport vector, initial conditions and other problem data in order to allow the existence and uniqueness proofs are discussed in each case. The proofs are based on the existence results for an approximating problem with certain Robin boundary conditions, indexed on a small parameter α . Appropriate estimates independent of α are established, and a passing to the limit technique, which is delicate due to the fact that the functional spaces in the approximating Robin problem depend on α , is performed. Results are given for both nondegenerate and degenerate cases.