Blow-up of the one-dimensional wave equation with quadratic spatial derivative nonlinearity

TL;DR

This paper studies the blow-up behavior of solutions to a one-dimensional wave equation with quadratic spatial derivative nonlinearity, establishing the existence and stability of certain self-similar blow-up solutions relevant in cosmology.

Contribution

It proves the non-existence of smooth self-similar blow-up solutions and constructs a family of generalized solutions with logarithmic growth, advancing spectral-theoretic methods for nonlinear wave equations.

Findings

Constructed a five-parameter family of self-similar solutions.

Proved asymptotic stability of these solutions.

Developed a robust spectral framework handling non-compact perturbations.

Abstract

We investigate the blow-up dynamics of smooth solutions to the one-dimensional wave equation with a quadratic spatial derivative nonlinearity, motivated by its applications in Effective Field Theory (EFT) in cosmology. Despite its relevance, explicit blow-up solutions for this equation have not been documented in the literature. In this work, we establish the non-existence of smooth, exact self-similar blow-up solutions and construct a five-parameter family of generalized self-similar solutions exhibiting logarithmic growth. Moreover, we prove the asymptotic stability of these blow-up solutions. Our proof tackles several significant challenges, including the non-self-adjoint nature of the linearized operator, the presence of unstable eigenvalues, and, most notably, the treatment of non-compact perturbations. By substantially advancing Donninger's spectral-theoretic framework, we develop a robust methodology that effectively handles non-compact perturbations. Key innovations include the incorporation of the Lorentz transformation in self-similar variables, an adaptation of the functional framework in [Merle-Raphael-Rodnianski-Szeftel, Invent.Math., 2022], and a novel resolvent estimate. This approach is general and robust, allowing for straightforward extensions to higher dimensions and applications to a wide range of nonlinear equations.