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

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