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

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 View the MathML sourceE1={u:ux=vxf(t)F(u),uy=vyf(t)F(u)} and View the MathML sourceE2={u:ux=a′(x)f(t)F(u),uy=b′(y)g(t)F(u)}, with f≠gf≠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.