On Stability of Type II Blow Up for the Critical Nonlinear Wave Equation in $\mathbb {R}^{3+1}$

We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation $ Box u = -u^5 $ on $ mathbb R^3+1$ constructed in Krieger,Schlag,and Tataru (2009) and Krieger and Schlag (2014) are stable along a co-dimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter $ lambda (t) = t^-1- nu $ is sufficiently close to the self-similar rate, i.e., $ nu >0$ is sufficiently small.

Format: Paperback / softback
Length: 267 pages
Publication date: 01 July 2021
Publisher: American Mathematical Society

The author demonstrates that the finite time type II blow-up solutions for the energy-critical nonlinear wave equation $Box u = -u^5$ on $mathbb R^3+1$ constructed in Krieger, Schlag, and Tataru (2009) and Krieger and Schlag (2014) are stable along a co-dimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter $lambda(t) = t^-1 - nu$ is sufficiently close to the self-similar rate, i.e., $nu > 0$ is sufficiently small. Our method is based on Fourier techniques adapted to time-dependent wave operators of the form $-partial_t^2 + partial_r^2 + frac 2r partial_r + V(lambda(t)r)$ for suitable monotone scaling parameters $lambda(t)$ and potentials $V(r)$ with a resonance at zero.

We begin by considering the equation $Box u = -u^5$ on $mathbb R^3+1$, which is an energy-critical nonlinear wave equation. In Krieger, Schlag, and Tataru (2009) and Krieger and Schlag (2014), the authors constructed finite time type II blow-up solutions for this equation. However, it is important to note that these solutions are not stable in the sense that they can undergo catastrophic blow-up in finite time.

To address this stability issue, the author proposes a new method based on Fourier techniques. The key idea is to adapt Fourier techniques to time-dependent wave operators of the form $-partial_t^2 + partial_r^2 + frac 2r partial_r + V(lambda(t)r)$, where $V(r)$ is a potential with a resonance at zero. The scaling parameter $lambda(t)$ is chosen to be close to the self-similar rate, which ensures the stability of the solutions.

The author then demonstrates the stability of the finite time type II blow-up solutions for the energy-critical nonlinear wave equation $Box u = -u^5$ on $mathbb R^3+1$ along a co-dimension three manifold of radial data perturbations in a suitable topology. The stability condition is that the scaling parameter $lambda(t)$ is sufficiently close to the self-similar rate, i.e., $nu > 0$ is sufficiently small.

The author's method is a significant breakthrough in the study of finite time type II blow-up solutions for energy-critical nonlinear wave equations. It provides a new tool for analyzing and understanding these solutions, which can have important applications in various fields, such as physics, mathematics, and engineering.

In conclusion, the author's work demonstrates that the finite time type II blow-up solutions for the energy-critical nonlinear wave equation $Box u = -u^5$ on $mathbb R^3+1$ constructed in Krieger, Schlag, and Tataru (2009) and Krieger and Schlag (2014) are stable along a co-dimension three manifold of radial data perturbations in a suitable topology, provided the scaling parameter $lambda(t)$ is sufficiently close to the self-similar rate, i.e., $nu > 0$ is sufficiently small. The author's method is based on Fourier techniques adapted to time-dependent wave operators, and it provides a new tool for analyzing and understanding these solutions.

Weight: 260g
ISBN-13: 9781470442996