O.D.E. type behavior of blow-up solutions of nonlinear heat equations

Abstract

We show that all global (in time and in space) and bounded solutions of the vector-valued equation <p align="center"> $\frac{\partial w}{\partial s}=\Delta w-\frac{1}{2}y \cdot \nabla w -\frac{w}{p-1}+|w|^{p-1}w$ <p align="left" class="times"> (where $w : \mathbb R^N\times \mathbb R \to \mathbb R^M, p> 1$ and $(N - 2)p < N + 2$) are independent of space and completely explicit. We then derive from this various uniform estimates and a uniform localization property for blow-up solutions of $\partial_t u=\Delta u + |u|^{p-1}u$.