Authors: Boris Andreianov; Noureddine Igbida
Addresses: Laboratoire de Mathématiques, CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25 030 Besançon Cedex, France. ' Institut de recherche XLIM, UMR-CNRS 6172, Université de Limoges, 123, Avenue Albert Thomas, 87060 Limoges, France
Abstract: We survey recent developments and give some new results concerning uniqueness of weak and renormalised solutions for degenerate parabolic problems of the form ut div(a0 (∇w) + F(w)) + f, u ∈ β(w) for a maximal monotone graph β, a Leray-Lions type nonlinearity a0, a continuous convection flux F, and an initial condition u|0 = u0. The main difficulty lies in taking boundary conditions into account. Here we consider Dirichlet or Neumann boundary conditions or the case of the problem in the whole space. We focus on techniques that are compatible with hyperbolic degeneracy, but here they serve to treat only the 'parabolic-elliptic aspects'. We revisit the derivation of entropy inequalities inside the domain and up to the boundary; technique of 'going to the boundary' in the Kato inequality for comparison of two solutions; uniqueness for renormalised solutions obtained via reduction to weak solutions. On several occasions we use the nonlinear semigroup theory.
Keywords: Stefan type problems; well-posedness; entropy inequlities; Dirichlet boundary conditions; Neumann boundary conditions; doubling of variables; Kato inequality; renormalised solutions; integral solutions; nonlinear semigroups; convection diffusion.
International Journal of Dynamical Systems and Differential Equations, 2012 Vol.4 No.1/2, pp.3 - 34
Available online: 26 Mar 2012 *Full-text access for editors Access for subscribers Purchase this article Comment on this article