Invariant sets and solutions to the two-dimensional nonlinear reaction–diffusion equations

Abstract

In this paper we utilize the method of invariant set to derive exact solutions of the two-dimensional nonlinear reaction–diffusion equations with source term. It is shown that there exists a class of reaction–diffusion equations which are invariant with respect to the sets E 1 = { u : u x = v x f ( t ) F ( u ) , u y = v y f ( t ) F ( u ) } and E 2 = { u : u x = a ′ ( x ) f ( t ) F ( u ) , u y = b ′ ( y ) g ( t ) F ( u ) } , with f ≠ g . As a result, we obtain exact solutions of the certain nonlinear reaction–diffusion equations. These solutions extend the well-known self-similar solutions and instantaneous source type solutions of the porous medium equation. The behavior to some solutions and the corresponding interfaces are also described.