Slow blow-up solutions for the H^1(R^3) critical focusing semi-linear wave equation in R^3

TL;DR

This paper constructs specific energy solutions to the energy-critical focusing wave equation in three dimensions that blow up precisely at the origin, decomposing into a rescaled soliton and radiation, using a novel renormalization approach.

Contribution

It introduces a new renormalization method to construct blow-up solutions for the energy-critical wave equation in R^3, demonstrating precise blow-up behavior.

Findings

Existence of solutions blowing up at a single point in space-time.

Decomposition into a rescaled soliton plus radiation.

Application of a novel renormalization technique.

Abstract

We prove the existence of energy solutions of the energy critical focusing wave equation in R^3 which blow up exactly at x=t=0. They decompose into a bulk term plus radiation term. The bulk is a rescaled version of the stationary "soliton" type solution of the NLW. The construction depends crucially on the renormalization procedure of the "soliton" which we introduced in our companion paper on the wave map problem.