On type I blow up formation for the critical NLW

TL;DR

This paper develops a weak evolution framework for the radial quintic focusing semilinear wave equation, demonstrating that it leads to type I blow-up in certain initial data scenarios, including known blow-up solutions.

Contribution

It introduces a new weak evolution concept that allows continuation past type II singularities, revealing type I blow-up formation for specific initial data.

Findings

Weak evolution concept enables continuation past singularities

Type I blow-up occurs for initial data near known blow-up solutions

Results apply to solutions with energy below or near the ground state

Abstract

We introduce a suitable concept of weak evolution in the context of the radial quintic focussing semilinear wave equation on $\mathbb{R}^{3+1}$, that is adapted to continuation past type II singularities. We show that the weak extension leads to type I singularity formation for initial data corresponding to: (i) the Kenig-Merle blow-up solutions with initial energy below the ground state and (ii) the Krieger-Nakanishi-Schlag blow-up solutions sitting initially near and "above" the ground state static solution.