Asymptotic Stability of a Stationary Solution of a Multidimensional Reaction–Diffusion Equation with a Discontinuous Source

Abstract

A two-dimensional reaction–diffusion equation in a medium with discontinuous characteristics is considered; the existence, local uniqueness, and asymptotic stability of its stationary solution, which has a large gradient at the interface, is proved. This paper continues the authors’ works concerning the existence and stability of solutions with internal transition layers of boundary value problems with discontinuous terms to multidimensional problems. The proof of the existence and stability of a solution is based on the method of upper and lower solutions. The methods of analysis proposed in this paper can be generalized to equations of arbitrary dimension of the spatial variables, as well as to more complex problems, e.g., problems for systems of equations. The results of this work can be used to develop numerical algorithms for solving stiff problems with discontinuous coefficients.