Focusing solutions for the $p$-Laplacian evolution equation

Abstract

We construct self-similar focusing solutions for the nonlinear parabolic equation $$ u_t=\Delta_pu=\text{div}(|\nabla u|^{p-2}\nabla u), $$ usually called the (evolution) $p$-Laplacian equation. We take the parameter $p>2$, so that finite propagation holds and free boundaries occur. We consider the problem posed in the whole space $R^n$ and work with nonnegative solutions. A self-similar solution is a solution $ u(x,t)$ that preserves its shape in time up to scaling. By focusing we mean that the solution vanishes for, say, $t<0$, in a ball of radius $s(t)$ centered at the origin and as $t\to 0$ we get $s(t)\to 0$, so that the hole disappears at $t=0$. We have $s(t)=c\,(-t)^\nu$ and the main point is the calculation of the anomalous exponent $\nu=\nu(n,p)$. The behaviour of the solution near $(0,0)$ is important in the regularity theory because when focusing occurs in dimension greater than $1$ a singularity appears. Hence, focusing solutions supply concrete bounds for the regularity of general solutions of the equation, which are presumably optimal. A characteristic of the focusing problem for the PLE is the fact that $\nu$ is not a monotone function of $p$ so that minimal regularity at the focusing is obtained for an intermediate $p$ in the interval $(2,\infty)$.