Invariant sets and solutions to higher-dimensional reaction–diffusion equations with source term

The invariant set and solutions of the two-dimensional reaction–diffusion equation with source term ut=A(u)uxx+B(u)uyy+C(u)ux2+D(u)uy2+Q(u), is discussed. It is shown that there exist a class of solutions to the equation which belong to the invariant set E0={u:ux=vxF(u),uy=vyF(u)}, where v is some smooth function of x and y and F is smooth function of u to be determined. The approach is also developed to deal with the N-dimensional reaction–diffusion equations with source term.