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

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 $x\to \hat{I}¾(t)$, where $\hat{I}¾:[0,\infty )\to {\mathbb{R}}^n$ is a given curve.