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

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^{-(