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

Given any $k \geq 2$ 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 (x, t) = \sum_{j = 1}^{k} {(- 1)}^{j - 1} μ_{j}^{- \frac{n - 2}{2}} U (\frac{x}{μ_{j}}) + o (1) as t \to + \infty,

where $U (y)$ is the standard soliton

U (y) = α_{n} {(\frac{1}{1 + | y |^{2}})}^{\frac{n - 2}{2}}

and

μ_{j} (t) = β_{j} t^{- α_{j}}, α_{j} = \frac{1}{2} ({(\frac{n - 2}{n - 6})}^{j - 1} - 1)

if $n \geq 7$ . For $n = 6$ , the rate of the $μ_{j} (t)$ is different and it is also discussed. Letting $δ_{0}$ be the Dirac mass, we have energy concentration of the form

e [u (\cdot, t)] - e [U] ⇀ (k - 1) S_{n} δ_{0} as t \to + \infty,

where $S_{n} = J (U)$ . The initial condition can be chosen radial and compactly supported. We establish the codimension $k + n (k - 1)$ stability of this phenomenon for perturbations of the initial condition that have space decay $u_{0} (x) = O (| x |^{- α})$ , $α > (n - 2) ∕ 2$ , which yields finite energy of the solution.

Keywords

energy critical heat equation, infinite time blow-up

Mathematical Subject Classification 2010

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

Vol. 14, No. 5, 2021

Manuel del Pino, Monica Musso and Juncheng Wei

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors