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

TL;DR

This paper analyzes a class of radial generalized nonlinear p-Laplacian evolution equations, deriving conservation laws and symmetries, and uses these to find exact solutions and understand their physical properties.

Contribution

It provides a comprehensive symmetry and conservation law analysis for these equations and constructs exact solutions, including transformations to linear equations.

Findings

All low-order conservation laws are derived.

The equations can be linearized via hodograph transformation in certain cases.

Exact solutions describing interfaces and Green's functions are obtained.

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.