Stable Self-Similar Blow-Up In Nonlinear Wave Equations With Quadratic Time-Derivative Nonlinearities
Jie Liu, Faiq Raees
TL;DR
This work analyzes finite-time singularity formation in two 1D nonlinear wave equations with quadratic derivative nonlinearities: $\partial_{tt}u-\partial_{xx}u=(\partial_t u)^2$ and $\partial_{tt}u-\partial_{xx}u=(\partial_t u)^2-(\partial_x u)^2$. It constructs a five-parameter family of generalized self-similar blow-up profiles with logarithmic corrections, proves there are no smooth exact self-similar solutions, and establishes asymptotic stability for key branches via a spectral framework that combines Lorentz transformations, hypergeometric reductions, and a Lyapunov–Perron nonlinear analysis. The analysis reveals a clear spectral gap: only symmetry-induced eigenvalues $0$ and $1$ can be unstable, with all other spectrum lying in the stable half-plane; this underpins robust nonlinear stability results. The results unify the blow-up mechanisms for time-derivative NLWs and provide a precise description of blow-up behavior inside backward light cones, including ODE-type and generalized self-similar blow-up regimes. These findings advance the understanding of derivative-type nonlinearities and offer a rigorous template for stability analyses of self-similar blow-up in related hyperbolic PDEs.
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.