Stable Non-Gaussian Diffusive Profiles

Abstract

We prove two stability results for the scale invariant solutions of the nonlinear heat equation ${\partial }_{t}u=\Delta u-|u{|}^{p-1}u$ with $1<p<1+\frac{2}{n}$, $n$ being the spatial dimension. The first result is that a small perturbation of a scale invariant solution vanishes as $t\to \infty$. The second result is global, with a positivity condition on the initial data.

12 pages, Latex