Abstract
In this thesis, we investigate the asymptotic behaviour of positive global unbounded solutions to the critical semilinear heat equation. We construct the first example of non-radial infinite-time blow up solution in a 3-dimensional bounded domain. This generalizes the non-radial case proved by Cortazar, Del Pino and Musso in [7] for dimension n ≥ 5 and the radial result for n=3 by Galaktionov and King [14].Our analysis starts by selecting a good ansatz, which encloses all the main asymptotic properties of the exact solution. We show that, after necessary improvements of the natural approximation, we get a sufficiently small error to start the second part of the proof.
Then, we produce a correct perturbation of the approximate solution by adapting the parabolic inner-outer gluing method developed in [7,8]. This consists in solving a weakly coupled system after suitably decomposing the problem near and far from the blow-up point. This approach is constructive and allows an accurate analysis of the asymptotic dynamics.
A fundamental feature and difficulty in the inner regime is the presence of a nonlocal operator that controls the second order asymptotic of the blow-up parameter. We show that such operator, similar to a half-fractional derivative, can be inverted but a loss of regularity in the parameter appears. We prove the invertibility of such operator using a Laplace transform type method combined with heat kernel estimates and we regain regularity of the parameters using smoothness of the solution in the outer regime.
For the unit ball, our construction works for any blow-up point sufficiently close to the center. In particular, we give a new proof of the infinite-time blow-up at the center of a ball, firstly proved in [14] using matched expansion techniques. Our construction applies to any domain under a natural analytic condition, given in terms of the Robin function of the domain.
| Date of Award | 28 Jun 2023 |
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| Original language | English |
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| Supervisor | Manuel Del Pino (Supervisor) & Monica Musso (Supervisor) |