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EXISTENCE AND STABILITY OF INFINITE TIME BUBBLE TOWERS IN THE ENERGY CRITICAL HEAT EQUATION
Abstract
We consider the energy-critical heat equation in ℝn for n ≥ 6 (Formula presented) which corresponds to the L2-gradient flow of the Sobolev-critical energy (Formula presented) 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, (Formula presented) where U(y) is the standard soliton (Formula presented) and (Formula presented) if n ≥ 7. For n = 6, the rate of the μj(t) is different and it is also discussed. Letting δ0 be the Dirac mass, we have energy concentration of the form (Formula presented) where Sn = J(U). The initial condition can be chosen radial and compactly supported. We establish the codimension k+n (k - l) stability of this phenomenon for perturbations of the initial condition that have space decay u0(x) = O(|x|−α), α > (n - 2)/2, which yields finite energy of the solution.