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 $${\mathrm{\partial }}_{t}v-\mathrm{\Delta }v-v^3=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^4\mathrm{.}$$ 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 $$\mathrm{\parallel }v(t){\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^4)}\approx \frac{\mathrm{\mid }\log (T-t){\mathrm{\mid }}^{\frac{2L}{2L-1}}}{(T-t{)}^L},L=1,2,\cdots \mathrm{.}$$ 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 $L\ge 2$.