A higher speed type II blow-up for the 4-d Fujita equation

TL;DR

This paper constructs higher speed type II blow-up solutions for the 4D energy critical heat equation, revealing new blow-up rates using the inner-outer gluing method.

Contribution

It demonstrates the existence of a family of type II blow-up solutions with specific blow-up rates for the 4D critical heat equation, advancing understanding of blow-up dynamics.

Findings

Existence of type II blow-up solutions with precise rates

Blow-up rate characterized by (T-t)^{-k}|ln(T-t)|^{2k/(2k-1)}

Application of inner-outer gluing method to construct solutions

Abstract

This paper is concerned with blow-up solutions of the 4-dimensional energy critical heat equation $u_t=\Delta u + u^3$. Our main result is to show that the existence of type II blowup solutions, and $\|u(\cdot,t)\|_\infty\sim(T-t)^{-k}|ln(T-t)|^{\frac{2k}{2k-1}}, (k=1,2,3\cdots)$.The inner-outer gluing method has been employed.