Stable Non-Gaussian Diffusive Profiles

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 $1<p<1+{2\over n}$, $n$ being the spatial dimension. The first result is that a small perturbation of a scale invariant solution vanishes as $t\rightarrow\infty$. The second result is global, with a positivity condition on the initial data.