Conditional Lie-Bäcklund symmetries and solutions to (n+1)-dimensional nonlinear diffusion equation

Abstract:
We discuss a class of (n+1)-dimensional nonlinear diffusion equation with source which arises in many physical situations. It is shown that some radially symmetric equations admit of second-order conditional Lie-Bäcklund symmetries. As a result, the corresponding solutions associated with the symmetries are obtained explicitly, or they are reduced to solve two-dimensional dynamical systems. Those solutions extend the known ones such as instantaneous source solutions of the porous medium equation with absorption term. The phenomena of extinction and blow up and behavior to many of the solutions are described.