We consider the energycritical heat equation in
${\mathbb{R}}^{n}$ for
$n\ge 6$
$$\left\{\begin{array}{c}{u}_{t}=\Delta u+u{}^{\frac{4}{n2}}u\phantom{\rule{1em}{0ex}}\text{in}{\mathbb{R}}^{n}\times \left(0,\infty \right),\phantom{\rule{1em}{0ex}}\hfill \\ u\left(\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}},0\right)={u}_{0}\phantom{\rule{1em}{0ex}}\text{in}{\mathbb{R}}^{n},\phantom{\rule{1em}{0ex}}\hfill \end{array}\right.$$
which corresponds to the
${L}^{2}$gradient
flow of the Sobolevcritical energy
$$J\left(u\right)={\int}_{{\mathbb{R}}^{n}}e\left[u\right],\phantom{\rule{1em}{0ex}}e\left[u\right]:=\frac{1}{2}\nabla u{}^{2}\frac{n2}{2n}u{}^{\frac{2n}{n2}}.$$
Given any
$k\ge 2$ we find
an initial condition
${u}_{0}$
that leads to signchanging solutions with
multiple blowup 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,
$$u\left(x,t\right)=\sum _{j=1}^{k}{\left(1\right)}^{j1}{\mu}_{j}^{\frac{n2}{2}}U\left(\frac{x}{{\mu}_{j}}\right)+o\left(1\right)\phantom{\rule{1em}{0ex}}\text{as}t\to +\infty ,$$
where
$U\left(y\right)$ is
the standard soliton
$$U\left(y\right)={\alpha}_{n}{\left(\frac{1}{1+y{}^{2}}\right)}^{\frac{n2}{2}}$$
and
$${\mu}_{j}\left(t\right)={\beta}_{j}{t}^{{\alpha}_{j}},\phantom{\rule{1em}{0ex}}{\alpha}_{j}=\frac{1}{2}\left(\phantom{\rule{0.17em}{0ex}}{\left(\frac{n2}{n6}\right)}^{\phantom{\rule{0.17em}{0ex}}j1}1\right)$$
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
$$e\left[u\left(\phantom{\rule{0.3em}{0ex}}\cdot \phantom{\rule{0.3em}{0ex}},t\right)\right]e\left[U\right]\rightharpoonup \left(k1\right){S}_{n}\phantom{\rule{0.3em}{0ex}}{\delta}_{0}\phantom{\rule{1em}{0ex}}\text{as}t\to +\infty ,$$
where
${S}_{n}=J\left(U\right)$.
The initial condition can be chosen radial and compactly supported. We establish the
codimension
$k+n\left(k1\right)$
stability of this phenomenon for perturbations of the initial condition that have space
decay
${u}_{0}\left(x\right)=O\left(x{}^{\alpha}\right)$,
$\alpha >\left(n2\right)\u22152$,
which yields finite energy of the solution.
