Stable Non-Gaussian Diffusive Profiles

We prove two stability results for the scale invariant solutions of the nonlinear heat equation ∂ t u = ∆ u −| u | p − 1 u with 1 < p < 1+ 2 n , n being the spatial dimension. The first result is that a small perturbation of a scale invariant solution vanishes as t → ∞ . The second result is global, with a positivity condition on the initial data.