We consider the energy-critical heat equation in
for
which corresponds to the
-gradient
flow of the Sobolev-critical energy
Given any
we find
an initial condition
that leads to sign-changing solutions with
multiple blow-up at a single point (tower of
bubbles) as
.
It has the form of a superposition with alternate signs of singularly scaled
Aubin–Talenti solitons,
where
is
the standard soliton
and
if
. For
, the rate of the
is different and it is
also discussed. Letting
be the Dirac mass, we have energy concentration of the form
where
.
The initial condition can be chosen radial and compactly supported. We establish the
codimension
stability of this phenomenon for perturbations of the initial condition that have space
decay
,
,
which yields finite energy of the solution.
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