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

Abstract

The invariant set and solutions of the two-dimensional reaction–diffusion equation with source term u t = A ( u ) u x x + B ( u ) u y y + C ( u ) u x 2 + D ( u ) u y 2 + Q ( u ) , is discussed. It is shown that there exist a class of solutions to the equation which belong to the invariant set E 0 = { u : u x = v x F ( u ) , u y = v y F ( 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.