Ivariant Sets And Exact Solution

It's a very important to obtain the exact solution of a partial different equation. In this paper, we use invariant sets to solve some nonlinear equations. It's a very effective approach. V.A.Galalkionov proposed a " nonlinear " extension to the ordinary scaling group, which is described by the invariance of the set So = {u : u_x = (1/x)F(u)}. The extension has been used to construct exact solution to some quasilinear evolution equations, KDV-type equation and higher order nonlinear equations. Qu and Estevez further extended the scaling group to more form and successfully to construct solutions to a number of nonlinear evolution equtions.In this paper, we furthe extend Galakionov's approach, which are described by the setE_o = {u : u_x = v_xF(u), u_y — v_yF(u)}.We use this approach to discuss two-dimensional reaction-diffusion equation with source termu_t = A(u)u_xx + B{u)u_yy + c(u)u_x² + D(u)u_y² + Q(u), and the generalized thin film equtionu_t = -div(A(u) Δ▽ + B(u) â–½uΔu + C(u)|â–½ u|2 â–½ u) + Q(u),where v is a smooth function of variables x, y and F is smooth function to be determined,â–½u = (u_x, u_y), Δu = u_xx + u_yy, A, B, C, D and Q are smooth functions of u. This approach also can be futher extended to discuss N-simensional reaction-diffusion equation. We can regard the application in the generalized thin film equation as extension of the results for 1+1-dimensional nonlinear eqution.