Vol. 14, No. 5, 2021

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
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
Milestones
Received: 2 October 2019
Revised: 3 December 2019
Accepted: 5 February 2020
Published: 22 August 2021
Authors
 Manuel del Pino Department of Mathematical Sciences University of Bath Bath United Kingdom Departamento de Ingeniería Matemática - CMM Universidad de Chile Santiago Chile Monica Musso Department of Mathematical Sciences University of Bath Bath United Kingdom Juncheng Wei Department of Mathematics University of British Columbia Vancouver, BC Canada