Infinite-time blow-up for the 3-dimensional energy-critical heat equation
Infinite-time blow-up for the 3-dimensional energy-critical heat equation
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DOI：
10.2140/apde.2020.13.215
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发表时间：
2017-05
期刊：
影响因子：
2.2
通讯作者：
M. Pino;M. Musso;Juncheng Wei
中科院分区：
文献类型：
--
作者：
M. Pino;M. Musso;Juncheng Wei
我们构造了三维能量临界热方程在时间上全局定义的无界正解。对于每个$\Gamma>；1$，我们找到了具有$\Lim\Limits_{r\to\inty}|x|^\Gamma u_0(X)>；的初始数据(不一定是径向对称的)。0$，如$t到$$u(\cdot，t)\|_\inty\sim t^{\Gamma-1\over 2}，\quad{\mbox{if}}\quad 1 2，\quad$$和$$u(\cdot，t)\|\inty\sim\sqrt{t}，(\ln t)^{-1}，\quad{\Mbox{if}\quad\Gamma=2.$$此外，我们还证明了这种无限时间爆破是协维稳定的。这种解决方案的存在是菲拉和金猜测的。
We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension three $$ u_t = \Delta u + u^5 , \quad {\mbox {in}} \quad \R^3 \times (0,\infty), \ \ u(x, 0)= u_0 (x)\inn \R^3. $$ For each $\gamma>1$ we find initial data (not necessarily radially symmetric) with $\lim\limits_{r \to \infty} |x|^\gamma u_0 (x) >0$ such that as $t \to \infty$ $$ \| u(\cdot ,t ) \|_\infty \sim t^{\gamma-1 \over 2} , \quad {\mbox {if}} \quad 1 2, \quad $$ and $$ \| u(\cdot , t)\|_\infty \sim \sqrt{t}\, (\ln t )^{-1} , \quad {\mbox {if}} \quad \gamma = 2. $$ Furthermore we show that this infinite time blow-up is co-dimensional one stable. The existence of such solutions was conjectured by Fila and King.