Stable Self-Similar Blow-Up In Nonlinear Wave Equations With Quadratic Time-Derivative Nonlinearities

TL;DR

This paper investigates the formation of singularities in one-dimensional nonlinear wave equations with quadratic time-derivative nonlinearities, classifying blow-up solutions and establishing their stability.

Contribution

It constructs and classifies a five-parameter family of generalized self-similar blow-up solutions and proves their asymptotic stability, providing a unified understanding of blow-up mechanisms.

Findings

No smooth exact self-similar profiles exist.

Generalized self-similar solutions exhibit logarithmic growth.

Asymptotic stability of blow-up solutions is established.

Abstract

We study singularity formation in two one-dimensional nonlinear wave models with quadratic time-derivative nonlinearities. The non-null model violates the null condition and typically develops finite-time blow-up; the null-form model is Lorentz-invariant and enjoys small-data global existence, yet still admits blow-up for large data. Building on our earlier work on spatial-derivative nonlinearities, we construct and classify a five-parameter family of generalized self-similar blow-up solutions that captures the observed dynamics. We prove that no smooth exact self-similar profiles exist, while the generalized self-similar solutions -exhibiting logarithmic growth- provide the correct blow-up description inside backward light cones. We further establish asymptotic stability for the relevant branches, including the ODE-type blow-up in both models. These results yield a coherent and unified picture of blow-up mechanisms in time-derivative nonlinear wave equations.