Small-Data Lifespan for a One-Dimensional Wave Equation with Mixed Characteristic-Time Derivative Source

Abstract

We study the lifespan of classical solutions to \[ v_{tt}-v_{xx}=|v_t+v_x|^m|v_t|^n, \qquad x\in\R,\quad t>0, \] where \(m>1\) and \(n>1\). For compactly supported data \((\eta\phi,\eta\psi)\), with \(\phi\in C_0^2(\R)\) and \(\psi\in C_0^1(\R)\), we prove the two-sided estimate \[ c\eta^{-(m+n-1)} \leq T(\eta) \leq C\eta^{-(m+n-1)} \] under the single one-sided assumption \(\psi-\phi'\geq0\) on \(\R\), the data being nontrivial. The lower bound is obtained from the characteristic integral system and holds without any sign restriction; the upper bound follows from a scalar superlinear inequality along a selected characteristic. A short argument shows that the sign assumption already forces \(\psi(x_0)+\phi'(x_0)>0\) at some point, so that no separate activation hypothesis is needed. We also show that the compatible cancellation condition \(\psi+\phi'\equiv0\) produces the global free wave \(v(x,t)=\eta\phi(x-t)\), and that this cancellation regime meets the sign assumption only for trivial data. The model therefore separates a cancellation regime from a finite-time amplification regime with an exactly determined lifespan scale.