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

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.