On the Stability of Type I Blow Up for the Energy Super Critical Heat Equation

On the Stability of Type I Blow Up for the Energy Super Critical Heat Equation Synopsis

The authors consider the energy super critical semilinear heat equation $\partial _{t}u=\Delta u u^{p}, x\in \mathbb{R}^3, p>5.$ The authors first revisit the construction of radially symmetric self similar solutions performed through an ode approach and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. They then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional nonradial 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 nonradial energy super critical settings.