Viscosity solutions of a class of degenerate quasilinear parabolic equations with Dirichlet boundary condition

• Published in: Nonlinear analysis Vol. 75; no. 7; pp. 3292 - 3312
• Main Authors: Ruan, W.H., Pao, C.V.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01-05-2012
• Subjects: Boundaries; Boundary conditions; Degenerate quasilinear parabolic equations; Dirichlet boundary condition; Dirichlet problem; Mathematical analysis; Mathematical models; Nonlinearity; Reaction diffusion equations; Viscosity; Viscosity solutions; Reaction diffusion equations; Degenerate quasilinear parabolic equations; Dirichlet boundary condition; Viscosity solutions
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2011.12.017

This paper is concerned with viscosity solutions for a class of degenerate quasilinear parabolic equations in a bounded domain with homogeneous Dirichlet boundary condition. The equation under consideration arises from a number of practical model problems including reaction–diffusion processes in a porous medium. The degeneracy of the problem appears on the boundary and possibly in the interior of the domain. The goal of this paper is to establish some comparison properties between viscosity upper and lower solutions and to show the existence of a continuous viscosity solution between them. An application of the above results is given to a porous-medium type of reaction–diffusion model which demonstrates some distinctive properties of the solution when compared with the corresponding semilinear problem.