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

Abstract: 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 this class, all local conservation laws of low-order and all Lie symmetries are derived. The physical meaning of the conservation laws is discussed, and one of the conservation laws is used to show that the nonlinear equation can be mapped invertibly into a linear equation by a hodograph transformation in certain cases. The symmetries are used to derive exact group-invariant solutions from solvable three-dimensional subgroups of the full symmetry group, which yields a direct reduction of the nonlinear equation to a quadrature. The physical and analytical properties of these exact solutions are explored, some of which describe moving interfaces and Green's functions.