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
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.
"Existence and stability of infinite time bubble towers in the energy critical heat equation." Anal. PDE 14 (5) 1557 - 1598, 2021. https://doi.org/10.2140/apde.2021.14.1557