Symmetry Of The Generalized Nonlinear Diffusion Equation

The nonlinear phenomena widely appear in almost all the scientific fields such as physics, chemistry, biology, society, economy and so on. With the development of science, researches on the nonlinear systems are making more and more progress. As a result, to deal with the nonlinear equations that describe the nonlinear systems becomes one of the main and hot topics for the researchers. It's well known that the symmetry reduction method related to group theory, such as Lie point symmetry method, generalized symmetry, conditional symmetry, generalized conditional symmetry and etc, is an effective tool to the study of the exact solutions for the nonlinear Partial Differential Equations.It is Zhdanov, Fokas and Liu who first presented the generalized conditional symmetry method, which is one of the most effective symmetry method. The method has been used to study the symmetry reduction and exact solutions of some nonlinear differential equations, the solutions here generally can't be obtained via the classical symmetry method or the conditional symmetry method.In this paper, the generalized conditional symmetry method, which can be considered as a generalization of the conditional symmetry method, is used to study the nonlinear diffusion equations u_t = [u^m(u_x)ⁿ]_x+P(u)u_x+Q(u) with power law diffusivity, which has quite a large number of physical applications. We obtain conditions under which the equations admit the second-order generalized conditional symmetries η = u_xx + H(u)u)x² + G(u)u_x + F(u) and η = u_xx + H(u)u)x² + G(u)(u_x)^2-n + F(u)(u_x)^1-n. A complete classification of the functional forms of the reaction coefficients and source terms is presented when the equation admits the generalized conditional symmetry reduction. Exact solutions to the resulting equations are constructed via the compatibility of the generalized conditional symmetry and the considered equation. These group invariant solutions, including extinction solution, blow-up solution and periodic solution, provide some important information to further study the equations.It's certain that the generalized conditional method can be also used to study the high-order diffusion-like equations, such as the fourth-order thin film equation or the fractional diffusion equation and their high-dimensional forms.