On Type I Blow-Up Formation for the Critical NLW

Abstract

We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation u "´u 5 on R 3`1 constructed in [28], [27] are stable along a co-dimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter λptq " t´1´ν is sufficiently close to the self-similar rate, i. e. ν ą 0 is sufficiently small. Our method is based on Fourier techniques adapted to time dependent wave operators of the forḿfor suitable monotone scaling parameters λptq and potentials Vprq with a resonance at zero.1991 Mathematics Subject Classification. 35L05, 35B40.

Stable blowup for wave equations in odd space dimensions

“…In the energy critical case, u T is the unique self-similar solution and it can be used to construct blowup solutions that diverge in the scale invariant norṁ H 1 × L 2 (R 3 ), cf. for example [30]. This behavior is referred to as type I and contrasted by type II blowup, where solutions stay bounded in the critical norm.…”

On Blowup in Supercritical Wave Equations

“…Many recent works focus on energy-critical equations where type II blowup solutions are studied that emerge from a dynamical rescaling of a soliton, e.g. [29,41,12,11,39,18,21,23,24,17,20,19,27], see also [42,38,36,37].…”

“…• The topology in which we consider the problem is optimal in the class of Sobolev spaces H k × H [29,41,12,11,39,18,21,23,24,17,20,19,27], see also [42,38,36,37]. In the supercritical case, blowup is typically self-similar.…”

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

“…The author and Schörkhuber [16,18,17] proved the asymptotic stability of the ODE blowup profile, but in the stronger topology H 2 × H 1 . We also mention the recent paper by Krieger and Wong [44] on continuation beyond type II singularities. Unfortunately, the impressive machinery [48,49,50,51,53,52] developed by Merle and Zaag for studying type I blowup is confined to energy-subcritical equations and does not apply here.…”