论文标题
临界指数的小分热方程中的无限时间气泡塔
Infinite time bubble towers in the fractional heat equation with critical exponent
论文作者
论文摘要
在本文中，我们考虑$ \ Mathbb {r}^n $中的分数加热方程，$ n> 6s，s \ in（0,1），$ \ begin {equation*} u_t = - （ - Δ） \ Mathbb {r}^n \ times \ Mathbb {r}。 \ end {equation*}我们通过在单个点上以多个爆炸的形式\ begin {equation {equation*}建立签名改变解决方案来构建一个为前向和向后问题的气泡塔类型解决方案。 u（x，t）=（1+o（1））\ sum_ {j = 1}^{k}（ - 1）^{j-1}μ_j（t）^{ - \ frac {n-2s} {2} {2}}}}} u \ left（\ frac {\ frac {x}} {x} {μ_j}} t \ to+\ infty，\ end {equation*}和带有多个爆炸的积极解决方案，单点\ begin \ begin {equination*} u（x，t）=（1+o（1））\ sum_ {j = 1}^{k}μ_j（t）^{ - \ frac {n-2s} {2}} {2}} u \ left（\ frac {x}} \ end {equation*}。这里$ k \ ge2 $是一个正整数，$$ u（y）=α_{n，s} \ left（\ frac {1} {1+| y |^2} \ right） | t |^{ - α_j}（1+o（1）） α_j= \ frac {1} {2S} \ left（\ frac {n-2s} {n-6s}} \ right）^{j-1} - \ frac {1} {2S} {2S}，\ end end {eark {earkation*}，对于某些某些积极的数字$β_j，j = 1，\ cdots，
In this paper, we consider the fractional heat equation with critical exponent in $\mathbb{R}^n$ for $n>6s,s\in(0,1),$ \begin{equation*} u_t=-(-Δ)^su+|u|^{\frac{4s}{n-2s}}u,\quad (x,t)\in \mathbb{R}^n\times\mathbb{R}. \end{equation*} We construct a bubble tower type solution both for the forward and backward problem by establishing the existence of the sign-changing solution with multiple blow-up at a single point with the form \begin{equation*} u(x,t)=(1+o(1))\sum_{j=1}^{k}(-1)^{j-1}μ_j(t)^{-\frac{n-2s}{2}}U\left(\frac{x}{μ_j(t)}\right) \quad\mbox{as}\quad t\to+\infty, \end{equation*} and the positive solution with multiple blow-up at a single point with the form \begin{equation*} u(x,t)=(1+o(1))\sum_{j=1}^{k}μ_j(t)^{-\frac{n-2s}{2}}U\left(\frac{x}{μ_j(t)}\right) \quad\mbox{as}\quad t\to-\infty, \end{equation*} respectively. Here $k\ge2$ is a positive integer, $$U(y)=α_{n,s}\left(\frac{1}{1+|y|^2}\right)^{\frac{n-2s}{2}},$$ and \begin{equation*} μ_j(t)=β_j |t|^{-α_j}(1+o(1))~\mathrm{as}~t\to\pm\infty, \quad α_j=\frac{1}{2s}\left(\frac{n-2s}{n-6s}\right)^{j-1}-\frac{1}{2s}, \end{equation*} for some certain positive numbers $β_j,j=1,\cdots,k.$
