Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation

Abstract

Abstract. We establish Strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a (time-dependent) self-similar potential. As an application we consider the critical wave equation and prove the asymptotic stability of the ODE blowup profile in the energy space.

Mode Stability of Self-Similar Wave Maps in Higher Dimensions

“…Based on numerics [3], ψ T is conjectured to describe the generic blow-up profile. The nonlinear asymptotic stability of ψ T was rigorously proved in [14,20,9], see also [17,15,18,19,16] for similar results related to other equations. In the case d ≥ 4, Bizoń and Biernat [1] discovered the explicit self-similar solution ψ T (t, r) = 2 arctan…”

Soliton resolution along a sequence of times for the focusing energy critical wave equation

“…These solutions are called type I blow up solutions. It is expected that any type I solution decomposes into a finite sum of explicit nonlinear objects related to this ODE blow-up, as in the 1-d case (see [59] and references therein), however very little is known in the energy-critical case (see [17] for a local study).…”

Dynamics of the focusing critical wave equation

“…In the energy-critical case there are numerical evidences that generic blow-up solutions behave like y 0 (t) see Bizoń, Chmaj and Tabor [3]. The stability of y 0 in light cones, in the energy topology was proved by Donninger [8]; see also previous results in stronger topology by Donninger and Schörkhuber [11].…”