Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations

TL;DR

This paper constructs viscosity solutions for a class of nonlinear singular diffusion equations, analyzing their properties, convergence, and energy monotonicity, including a generalization of Struwe's monotonicity formula.

Contribution

It introduces a regularization scheme to construct solutions and proves a new monotonicity formula for weighted energy in nonlinear diffusion equations.

Findings

Solutions converge as p approaches 1.

Established comparison principles for the solutions.

Proved a generalized energy monotonicity formula.

Abstract

We construct viscosity solutions to the nonlinear evolution equation \eqref{p} below which generalizes the motion of level sets by mean curvature (the latter corresponds to the case $p = 1$) using the regularization scheme as in \cite{ES1} and \cite{SZ}. The pointwise properties of such solutions, namely the comparison principles, convergence of solutions as $p\to 1$, large-time behavior and unweighted energy monotonicity are studied. We also prove a notable monotonicity formula for the weighted energy, thus generalizing Struwe's famous monotonicity formula for the heat equation ($p =2$).