#### Vol. 14, No. 5, 2021

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Existence and stability of infinite time bubble towers in the energy critical heat equation

### Manuel del Pino, Monica Musso and Juncheng Wei

Vol. 14 (2021), No. 5, 1557–1598
##### Abstract

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

$J\left(u\right)={\int }_{{ℝ}^{n}}e\left[u\right],\phantom{\rule{1em}{0ex}}e\left[u\right]:=\frac{1}{2}|\nabla u{|}^{2}-\frac{n-2}{2n}|u{|}^{\frac{2n}{n-2}}.$

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,

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

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.

##### Keywords
energy critical heat equation, infinite time blow-up
Primary: 35K58
Secondary: 35B44