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

We consider the energy critical heat equation in Rⁿ for n≥ 7 \ aligned uₜ & = Δu+ |u|^ 4n-2u in \ Rⁿ × (0, ∞), \\ u(·,0) & = u₀ \ in \ Rⁿ, aligned. which corresponds to the L²-gradient flow of the Sobolev-critical energy J(u) = ∫Rⁿ e[u] , e[u] := 12 |∇ u|² - n-2/2n |u|2n/n-2. Given any k≥ 2 we find an initial condition u₀ that leads to sign-changing solutions with { multiple blow-up at a single point} (tower of bubbles) as t→ +∞. It has the form of a superposition with alternate signs of singularly scaled { Aubin-Talenti solitons}, u(x,t) = ∑ⱼ₌₁ᵏ (-1)ʲ⁻¹ μⱼ^- n-22 U ( x/μⱼ )\, +\, o(1) t→ +∞ where $U(y)$ is the standard soliton U(y) = % (n(n-2))^ 1n-2 αₙ ( 11+|y|²)^n-22 and μⱼ(t) = βⱼ t- αⱼ, αⱼ = 12 ( \, ( n-2/n-6)ʲ⁻¹ -1 ). Letting δ₀ the Dirac mass, we have energy concentration of the form e[ u(·, t)]- e[U] (k-1) Sₙ\,δ₀ t→ +∞ where Sₙ=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 u₀(x) =O( |x|-α), α> n-22, which yields finite energy of the solution.