Nonlinear Diffusion Equation Conditions Of The Lie-b (?) Backlund Symmetry And Invariant Subspaces

The nonlinear diffusion equations can be used to describe various diffu-sion process, including heat conduction in solid, the cooperation and com-petition of biological population, a.nd spread of infectious disease. The clas-sification and symmetry reductions of the nonlinear diffusion equations are discussed by using the conditional Lie-Backlund symmetry method.The conditional Lie-Backlund symmetry method can be regarded as a natural generalization of the nonclassical symmetry method in a similar way to how the Lie-Backlund symmetry method is a generalization of Lie's classical symmetry method. Therefore, the procedure for computing the conditional Lie-Backlund symmetry is about the same as for the nonclassical symmetry method. The most important point is to give the form of the conditional Lie-Backlund symmetry presumably. The conditional Lie-Backlund symmetries with the characteristic (?)are used to classify four types of nonlinear diffusion equations, including inhomoge-neous nonlinear diffusion equations, reaction-diffusion-convection equations, generalized porous medium equations and radially symmetric version of non-linear diffusion equations. That is, the unknown functions in the equations and the corresponding conditional Lie-Backlund symmetries are determined.Equation ut= F[u] admit the nonlinear conditional Lie-Backlund sym-metryηif and only if equation Ut= E[u] admit the linear conditional Lie-Backlund symmetryσ=ul.+a1(·)u(l-1)+…+al(·)u, where ut=E[u] can be transformed by v=f(u) from ut=F[u]. Equation Ut= E[U] ad-mit the linear conditional Lie-Backlund symmetryσis equivalent to say that the solution space of the linear ordinary differential equationσ=0 is exactly the invariant subspace of the operator E[u], that is, the conditional Lie-Backlund symmetry method can provide symmetry interpretation for the invariant subspace.The functional generalized separable solutions of the obtained equations from the classification are constructed due to the compatibility ofη=0 and ut= F[u], which can also be obtained from the transformationυ=f(u) and the generalized separable solutions determined on the invariant subspace ofυt= E[υ]. These solutions can describe singularities of blow-up and extin-guish, and the large time behavior, and etc. The undetermined functions in the obtained solutions satisfy the finite-dimensional dynamical system.