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

Abstract

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

$$\left\{\begin{array}{c}{u}_t=\mathrm{\Delta }u+|u{|}^{\frac{4}{n-2}}u\text{ in }{ℝ}^n\times \left(0,\infty \right),\hfill \\ u\left(\cdot ,0\right)=u_0\text{ in }{ℝ}^n,\hfill \end{array}\right.$$

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

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

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,

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

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

$${\mu }_j\left(t\right)={\beta }_j{t}^{-{\alpha }_j},{\alpha }_j=\frac{1}{2}\left({\left(\frac{n-2}{n-6}\right)}^{j-1}-1\right)$$

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

$$e\left[u\left(\cdot ,t\right)\right]-e\left[U\right]\rightharpoonup \left(k-1\right)S_n{\delta }_0\text{ as }t\to +\infty ,$$

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 $u_0\left(x\right)=O\left(|x{|}^{-\alpha }\right)$, , which yields finite energy of the solution.