On the stability of type I blow up for the energy super critical heat equation

Abstract

We consider the energy super critical semilinear heat equation \[\partial_t u=\Delta u+u^{p}, \ \ x\in \mathbb R^3, \ \ p>5.\] We first revisit the construction of radially symmetric backward self similar solutions and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. We then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional non radial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self similar blow up in non radial energy super critical settings.