论文标题
新型II型有限时间爆炸，用于能量超临界热方程
New type II Finite time blow-up for the energy supercritical heat equation
论文作者
论文摘要
我们考虑使用$（n-3）$ - th sobolev endent \ begin {equation*} \ begin {case} u_t =ΔU+u^{3}，〜＆\ mbox {in}ω\ times（times（0）， \partialΩ\times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } Ω, \end{cases} \end{equation*} where $5\leq n\leq 7$, $Ω=\R^n$ or $Ω\subset \R^n$ is a smooth, bounded domain enjoying special symmetries.我们构建II型有限时间爆破解决方案$ u（x，t）$，沿$（n-4）$ - 尺寸{\ em缩小球}以$ω$进行的奇异性。更确切地说，在领先顺序上，解决方案$ u（x，t）$属于较大的表单$$ u（x，t）\ aid \ la^{ - 1}（t）\ frac {2 \ sqrt {2}} {1+ \ left | \ frac {（r，z） - （ξ_r（t），ξ_z（t），ξ_z（t））} {\ la（t）} {\ la（t）}}} \ right |^2} $ the $ r = \ sqrt {x_1^2+\ cdots+x_ {n-3}^2} $，$ z =（x__ {n-2}，x_ {n-1}，x_n），x_n）$，带有$ x =（x_1，x_1，\ cdots，x_n）\ inch $。此外，奇点位置$$（ξ_r（t），ξ_z（t））\ sim（\ sqrt {2（n-4）（n-4）（t-t）}，Z_0） \ frac {t-t} {| \ log（t-t-t）|^2}〜\ mbox {as} 〜t \ nearlrowt。$$这是抛物线设置中的全新现象。
We consider the energy supercritical heat equation with the $(n-3)$-th Sobolev exponent \begin{equation*} \begin{cases} u_t=Δu+u^{3},~&\mbox{ in } Ω\times (0,T),\\ u(x,t)=u|_{\partialΩ},~&\mbox{ on } \partialΩ\times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } Ω, \end{cases} \end{equation*} where $5\leq n\leq 7$, $Ω=\R^n$ or $Ω\subset \R^n$ is a smooth, bounded domain enjoying special symmetries. We construct type II finite time blow-up solution $u(x,t)$ with the singularity taking place along an $(n-4)$-dimensional {\em shrinking sphere} in $Ω$. More precisely, at leading order, the solution $u(x,t)$ is of the sharply scaled form $$u(x,t)\approx \la^{-1}(t)\frac{2\sqrt{2}}{1+\left|\frac{(r,z)-(ξ_r(t),ξ_z(t))}{\la(t)}\right|^2}$$ where $r=\sqrt{x_1^2+\cdots+x_{n-3}^2}$, $z=(x_{n-2},x_{n-1},x_n)$ with $x=(x_1,\cdots,x_n)\inΩ$. Moreover, the singularity location $$(ξ_r(t),ξ_z(t))\sim (\sqrt{2(n-4)(T-t)},z_0)~\mbox{ as }~t\nearrow T,$$ for some fixed $z_0$, and the blow-up rate $$\la(t)\sim \frac{T-t}{|\log(T-t)|^2}~\mbox{ as }~t\nearrow T.$$ This is a completely new phenomenon in the parabolic setting.
