论文标题
对于四维能量临界半线性热方程的缓慢吹溶液
A slow blow up solution for the four dimensional energy critical semi linear heat equation
论文作者
论文摘要
我们考虑能量临界四维半线性热方程\ [\ partial_ {t} v-ΔV-v-v^{3} = 0，\ quad（t，x）\ in \ mathbb {r} \ times \ times \ times \ mathbb {r}^4。 \] Filippas等人的正式计算。 （R.Soc。Lond。Proc。2000）猜想存在具有各种爆炸率的II型爆炸解决方案\ [\ | v（t）\ | _ {l^\ iftty（\ Mathbb {r}^4）} \ aid \ frac {| \ log（t-t）|^{\ frac {2l} {2l-1}}}} {（t-t）^l}，\ quad l = 1,2，\ cdots。\ cdots。\ \] schweyer（J.funct。Anal。2012）严格地构建了II型bubl bubl solution for Case $ l = 1 $。在本文中，我们显示了$ L = 2 $的II型爆破解决方案的存在。可以将这里的方法推广以处理所有情况$ l \ geq 2 $。
We consider the energy critical four dimensional semi-linear heat equation \[ \partial_{t}v-Δv-v^{3}=0, \quad(t,x)\in \mathbb{R}\times \mathbb{R}^4. \] 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 $L\geq 2$.
