论文标题
关于尺寸四的Fila-King猜想
On Fila-King Conjecture in Dimension Four
论文作者
论文摘要
我们考虑以下四维能量临界热方程\ begin {equation*} \ begin {case} u_t =ΔU+u^{3}〜＆\ mbox {in}〜{ {\ Mathbb r}^4。 \ end {cases} \ end {equation*} 我们构建一个带有爆炸率$ \ |的积极的无限时间爆破解决方案$ u（x，t）$ u（\ cdot，t）\ | _ {l^\ infty（{\ mathbb r}^4）} \ sim \ ln t $ as $ t \ to \ to \ infty $，并显示了无限时间爆炸的稳定性。这给出了Fila和King \ Cite [猜想1.1] {Filaking12}的严格证明。
We consider the following Cauchy problem for the four-dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=Δu+u^{3} ~&\mbox{ in }~ {\mathbb R}^4 \times (0,\infty),\\ u(x,0)=u_0(x) ~&\mbox{ in }~ {\mathbb R}^4. \end{cases} \end{equation*} We construct a positive infinite time blow-up solution $u(x,t)$ with the blow-up rate $ \| u(\cdot,t)\|_{L^\infty({\mathbb R}^4)} \sim \ln t$ as $t\to \infty$ and show the stability of the infinite time blow-up. This gives a rigorous proof of a conjecture by Fila and King \cite[Conjecture 1.1]{filaking12}.
