We consider the energy-critical heat equation in ${ℝ}^n$ for $n\ge 6$

which corresponds to the $L^2$-gradient flow of the Sobolev-critical energy

Given any $k\ge 2$ we find an initial condition $u_0$ that leads to sign-changing solutions with multiple blow-up at a single point (tower of bubbles) as $t\to +\infty$. It has the form of a superposition with alternate signs of singularly scaled Aubin–Talenti solitons,

where $U\left(y\right)$ is the standard soliton

and

if $n\ge 7$. For $n=6$, the rate of the ${\mu }_j\left(t\right)$ is different and it is also discussed. Letting ${\delta }_0$ be the Dirac mass, we have energy concentration of the form

where $S_n=J\left(U\right)$. The initial condition can be chosen radial and compactly supported. We establish the codimension $k+n\left(k-1\right)$ stability of this phenomenon for perturbations of the initial condition that have space decay $u_0\left(x\right)=O\left(|x{|}^{-\alpha }\right)$, $\alpha >\left(n-2\right)∕2$, which yields finite energy of the solution.

我们考虑能量临界热方程 ${ℝ}^n$ 对于 $n\ge 6$

这对应于 ${升}^2$-Sobolev 临界能量的梯度流

给定任何 $克\ge 2$ 我们找到一个初始条件 ${你}_0$这导致在单个点（气泡塔）处多次爆炸的符号改变解决方案$吨\to +\infty$. 它具有叠加的形式，具有奇异缩放的Aubin-Talenti 孤子的交替符号，

哪里 $你\left(是\right)$ 是标准孤子

和

如果 $n\ge 7$. 对于$n=6$, 的速率 ${\mu }_j\left(吨\right)$是不同的，它也被讨论。出租${\delta }_0$ 是狄拉克质量，我们有形式的能量集中

哪里 ${秒}_n=J\left(你\right)$. 初始条件可以选择径向和紧凑支撑。我们建立代码$克+n\left(克-1\right)$ 对于具有空间衰减的初始条件的扰动，这种现象的稳定性 ${你}_0\left(\times \right)=哦\left(|\times {|}^{-\alpha }\right)$, $\alpha >\left(n-2\right)∕2$，这会产生解的有限能量。