## Abstract

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

$$\left\{\begin{array}{c}{u}_{t}=\mathrm{\Delta}u+|u{|}^{\frac{4}{n-2}}u\phantom{\rule{1em}{0ex}}\text{in}{\mathbb{R}}^{n}\times \left(0,\infty \right),\phantom{\rule{1em}{0ex}}\hfill \\ u\left(\cdot ,0\right)={u}_{0}\phantom{\rule{1em}{0ex}}\text{in}{\mathbb{R}}^{n},\phantom{\rule{1em}{0ex}}\hfill \end{array}\right.$$

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

$$J(u)={\displaystyle {\int}_{{\mathbb{R}}^{n}}{}_{}e[u],\text{\hspace{1em}}e[u]:=\frac{1}{2}|\nabla u{|}^{2}-\frac{n-2}{2n}|u{|}^{\frac{2n}{n-2}}}.$$

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*,

$$u\left(x,t\right)=\sum _{j=1}^{k}{\left(-1\right)}^{j-1}{\mu}_{j}^{-\frac{n-2}{2}}U\left(\frac{x}{{\mu}_{j}}\right)+o\left(1\right)\phantom{\rule{1em}{0ex}}\text{as}t\to +\infty ,$$

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

$$U\left(y\right)={\alpha}_{n}{\left(\frac{1}{1+|y{|}^{2}}\right)}^{\frac{n-2}{2}}$$

and

$${\mu}_{j}\left(t\right)={\beta}_{j}{t}^{-{\alpha}_{j}},\phantom{\rule{1em}{0ex}}{\alpha}_{j}=\frac{1}{2}\left(\phantom{\rule{-0.17em}{0ex}}{\left(\frac{n-2}{n-6}\right)}^{\phantom{\rule{-0.17em}{0ex}}j-1}-1\right)$$

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

$$e\left[u\left(\cdot ,t\right)\right]-e\left[U\right]\rightharpoonup \left(k-1\right){S}_{n}{\delta}_{0}\phantom{\rule{1em}{0ex}}\text{as}t\to +\infty ,$$

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 ${u}_{0}\left(x\right)=O\left(|x{|}^{-\alpha}\right)$, $\alpha >\left(n-2\right)\u22152$, which yields finite energy of the solution.

## Citation

Manuel del Pino. Monica Musso. Juncheng Wei. "Existence and stability of infinite time bubble towers in the energy critical heat equation." Anal. PDE 14 (5) 1557 - 1598, 2021. https://doi.org/10.2140/apde.2021.14.1557

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