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

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 not equal 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. (C) 2009 Elsevier Ltd. All rights reserved.