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

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.