Conservation laws and symmetries of radial generalized nonlinear $p$-Laplacian evolution equations

摘要

A class of generalized nonlinear p-Laplacian evolution equations is studied.These equations model radial diffusion-reaction processes in $n\geq 1$dimensions, where the diffusivity depends on the gradient of the flow. For thisclass, all local conservation laws of low-order and all Lie symmetries arederived. The physical meaning of the conservation laws is discussed, and one ofthe conservation laws is used to show that the nonlinear equation can be mappedinvertibly into a linear equation by a hodograph transformation in certaincases. The symmetries are used to derive exact group-invariant solutions fromsolvable three-dimensional subgroups of the full symmetry group, which yields adirect reduction of the nonlinear equation to a quadrature. The physical andanalytical properties of these exact solutions are explored, some of whichdescribe moving interfaces and Green's functions.