A slow blow up solution for the four dimensional energy critical semi linear heat equation

TL;DR

This paper proves the existence of a specific type II blow-up solution for the four-dimensional energy critical semi-linear heat equation, extending previous work to the case where L=2, with potential for generalization.

Contribution

It establishes the existence of a new type II blow-up solution for L=2, advancing understanding of blow-up behaviors in critical semi-linear heat equations.

Findings

Constructed a type II blow-up solution for L=2.

Method can be extended to all L≥2.

Supports conjectures on blow-up rate diversity.

Abstract

We consider the energy critical four dimensional semi-linear heat equation \[ \partial_{t}v-\Delta v-v^{3}=0, \quad(t,x)\in \mathbb{R}\times \mathbb{R}^4. \] Formal computation of Filippas et al. (R. Soc. Lond. Proc. 2000) conjectures the existence of a sequence of type II blow-up solutions with various blow-up rates \[ \|v(t)\|_{L^\infty(\mathbb{R}^4)}\approx \frac{|\log(T-t)|^{\frac{2L}{2L-1}}}{(T-t)^L} ,\quad L=1,2,\cdots.\] Schweyer (J. Funct. Anal. 2012) rigorously constructs a type II blow-up solution for the case $L=1$. In this paper, we show the existence of type II blow-up solution for $L=2$. The method here could be generalized to deal with all the cases $L\geq 2$.