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

Abstract

We consider the energy critical heat equation in ${\mathbb{R}}^n$ for $n\ge 7$ $$\{\begin{array}{rl}{u}_t& =\Delta u+|u{|}^{\frac{4}{n-2}}uin{\mathbb{R}}^n\times (0,\infty ),\\ u(\cdot ,0)& =u_{0}in{\mathbb{R}}^n,\end{array}$$ which corresponds to the $L^2$-gradient flow of the Sobolev-critical energy $$J(u)={\int }_{{\mathbb{R}}^n}e[u],e[u]:=\frac{1}{2}|\nabla u{|}^2-\frac{n-2}{2n}|u{|}^{\frac{2n}{n-2}}.$$ Given any $k\ge 2$ we find an initial condition $u_0$ that leads to sign-changing solutions with {\em 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 {\em Aubin-Talenti solitons}, $$u(x,t)=\sum _{j=1}^k(-1{)}^{j-1}{\mu }_j^{-\frac{n-2}{2}}U\left(\frac{x}{{\mu }_j}\right)+o(1)ast\to +\infty$$ where $U(y)$ is the standard soliton $U(y)=$ and $${\mu }_j(t)={\beta }_j{t}^{-{\alpha }_j},{\alpha }_j=\frac{1}{2}({\left(\frac{n-2}{n-6}\right)}^{j-1}-1).$$ Letting ${\delta }_0$ the Dirac mass, we have energy concentration of the form $$e[u(\cdot ,t)]-e[U]\rightharpoonup (k-1)S_n{\delta }_{0}ast\to +\infty$$ 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 $u_0(x)=O(|x{|}^{-\alpha })$, $\alpha >\frac{n-2}{2}$, which yields finite energy of the solution.

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Cited by in corpus (1)

• New type II Finite time blow-up for the energy supercritical heat equation