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

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 $L\geq 2$.