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On self-similar blowup for supercritical dispersive PDEs
Abstract
Numerical simulations of different types of supercritical evolution equations show certain degree of universality when it comes to formation of singularities. Specifically, it appears that the generic blowup behavior and threshold for blowup phenomena are both governed by self-similar solutions. In this talk, we explore these observations in the context of the focusing cubic wave equation in the energy-supercritical regime, d≥5. We begin by reviewing the results leading to a complete proof of the non-radial stability of the so-called ODE blowup profile. Next, we present what appears to be the only nontrivial self-similar solution known in closed form. We then show numerical evidence suggesting that this solution acts as a generic attractor within the threshold for the ODE blowup. Finally, as the first step toward rigorously showing this observation, we outline our proof of the non-radial co-dimension one stability of this solution. At the end we will comment on the analogous results/conjectures for other supercritical dispersive models. The talk is based on joint works with Maciej Maliborski (Vienna) and Birgit Schörkhuber (Innsbruck).
