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

Source abstract

We consider the energy super critical semilinear heat equation ∂ t u = Δ u + u p , x ∈ R 3 , p > 5. $$\begin{equation*} \partial _tu=\Delta u+u^{p}, \ \ x\in \Bbb R^3, \ \ p>5. \end{equation*}$$ We first revisit the construction of radially symmetric self similar solutions performed through an ode approach in Troy (1987), Budd and Qi (1989), and propose a bifurcation type argument suggested in Biernat and Bizon (2011) 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.