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 ((ηφ,ηψ)), with (φ\in C_0^2(\R)) and (ψ\in C_0^1(\R)), we prove the two-sided estimate [ cη^{-(m+n-1)} \leq T(η) \leq Cη^{-(m+n-1)} ] under the single one-sided assumption (ψ-φ'\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 (ψ(x_0)+φ'(x_0)>0) at some point, so that no separate activation hypothesis is needed. We also show that the compatible cancellation condition (ψ+φ'\equiv0) produces the global free wave (v(x,t)=ηφ(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.