Blow-up of the one-dimensional wave equation with quadratic spatial derivative nonlinearity
Tej-eddine Ghoul, Jie Liu, Nader Masmoudi
TL;DR
This work analyzes the one-dimensional wave equation with a quadratic derivative nonlinearity $u_{tt}-u_{xx}=(u_x)^2$ arising in cosmological EFT. It proves there are no smooth exact self-similar blow-ups and constructs a five-parameter family of generalized self-similar blow-up solutions with logarithmic growth, together with a proof of their asymptotic stability. The authors develop a robust spectral-theoretic framework that handles non-self-adjoint operators and non-compact perturbations, leveraging a Lorentz-transform in self-similar variables, a refined operator decomposition, and direct resolvent estimates. These innovations yield linear decay on the stable subspace and nonlinear stability via a Lyapunov-Perron construction, with potential extensions to higher dimensions and broader nonlinear models. The results provide rigorous insight into blow-up dynamics relevant to EFT cosmology and establish methods applicable to a wider class of nonlinear wave equations exhibiting self-similar blow-up behavior.
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.