Stable Non-Gaussian Diffusive Profiles

We prove two stability results for the scale invariant solutions of the nonlinear heat equation ${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u-\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ 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 $t\to \mathrm{\infty }$. The second result is global, with a positivity condition on the initial data.