Type II Blow Up for the Four Dimensional Energy Critical Semi Linear Heat Equation

TL;DR

This paper constructs and characterizes type II finite-time blow-up solutions for the four-dimensional energy-critical semi-linear heat equation, revealing universal singularity profiles and blow-up rates using advanced energy methods.

Contribution

It introduces the first known type II blow-up solutions for this equation and provides a detailed description of their singularity formation and blow-up dynamics.

Findings

Existence of type II finite-time blow-up solutions.

Universal bubble concentration in the critical topology.

Precise blow-up rate with logarithmic correction.

Abstract

We consider the energy critical four dimensional semi linear heat equation \partial tu-\Deltau-u3 = 0. We show the existence of type II finite time blow up solutions and give a sharp description of the corresponding singularity formation. These solutions concentrate a universal bubble of energy in the critical topology u(t,r)-1/{\lambda} Q(r/{\lambda})\rightarrow u* in $\dot{H}^1$ where the blow up profile is given by the Talenti Aubin soliton Q(r)= 1/(1 +r^2/8) and with speed {\lambda}(t) ~(T-t)/|log(T - t)|^2 as t\rightarrowT. Our approach uses a robust energy method approach developped for the study of geometrical dispersive problems, and lies in the continuation of the study of the energy critical harmonic heat flow and the energy critical four dimensional wave equation.