On strongly anisotropic type I blow up

Abstract

We consider the energy super critical 4 dimensional semilinear heat equation $$\partial_tu=\Delta u+|u|^{p-1}u, \ \ x\in \Bbb R^4, \ \ p>5.$$ Let $\Phi(r)$ be a three dimensional radial self similar solution for the three supercritical probmem as exhibited and studied in \cite{CRS}. We show the finite codimensional transversal stability of the corresponding blow up solution by exhibiting a manifold of finite energy blow up solutions of the four dimensional problem with cylindrical symmetry which blows up as $$u(t,x)\sim \frac{1}{(T-t)^{\frac{1}{p-1}}}U(t,Y), \ \ Y=\frac{x}{\sqrt{T-t}}$$ with the profile $U$ given to leading order by $$U(t,Y)\sim\frac{1}{(1+b(t)z^2)^{\frac 1{p-1}}}\Phi\left(\frac{r}{\sqrt{1+b(t)z^2}}\right), \ \ Y=(r,z), \ \ b(t)=\frac{c}{|\log(T-t)|}$$ corresponding to a constant profile $\Phi(r)$ in the $z$ direction reconnected to zero along the moving free boundary $|z(t)|\sim \frac{1}{\sqrt{b}}\sim \sqrt{|\log (T-t)|}.$ Our analysis revisits the stability analysis of the self similar ODE blow up \cite{BK, MZduke,MZgaffa} and combines it with the study of the Type I self similar blow up \cite{CRS}. This provides a robust canonical framework for the construction of strongly anisotropic blow up bubbles.