Separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source

Abstract

A solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q( u)= φ( x)+ ψ( t) for some single variable functions, q, φ and ψ. In this paper, the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source. We obtain a complete list of canonical forms for such equations which admit the functionally separable solutions. As a result, we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source. The behavior and blow-up properties of some solutions are described.