Moving gradient singularity for the evolutionary $p$-Laplace equation

TL;DR

This paper constructs solutions to the evolutionary p-Laplace equation for p>n that feature a moving gradient singularity along a specified curve, advancing understanding of singularity behavior in nonlinear PDEs.

Contribution

It introduces a novel method to create solutions with a moving gradient singularity in the evolutionary p-Laplace equation for p>n, highlighting dynamic singularity phenomena.

Findings

Existence of solutions with moving gradient singularities

Singularities follow a prescribed curve over time

Enhanced understanding of singularity dynamics in nonlinear PDEs

Abstract

We consider the evolutionary $p$-Laplace equation in $\mathbb{R}^n$. For $p>n$, we construct a solution $u$ with a moving gradient singularity in the sense that $|\nabla u(x,t)|\to \infty$ for each $t$ as $x\to\xi(t)$, where $\xi:[0,\infty)\to\mathbb{R}^n$ is a given curve.