Characterize lifespan behavior without the one-sided sign condition

Determine the lifespan and possible blow-up behavior for the one-dimensional semilinear wave equation \(v_{tt}-v_{xx}=|v_t+v_x|^m|v_t|^n\) when the initial-data condition \(\psi-\phi'\geq0\) is removed, including whether a comparable lower bound along the selected characteristic can still be obtained.

Background

The paper’s upper lifespan estimate relies on ψ−ϕ′≥0\psi-\phi'\geq0, which propagates nonnegativity of the characteristic component Q=vt−vxQ=v_t-v_x. This sign allows the factor ∣vt∣n=∣(P+Q)/2∣n|v_t|^n=|(P+Q)/2|^n to be bounded below in terms of the growing component P=vt+vxP=v_t+v_x along a characteristic. Without that condition, the source may not admit the same scalar lower bound, and the paper does not resolve the resulting lifespan and blow-up questions.

References

Two further questions are what happens when eq:sharp-Q is dropped, in which case the source need not admit a lower bound in terms of P alone along the selected characteristic, and whether the blow-up mechanism identified here is compatible with a description of the blow-up curve in the spirit of .

— Small-Data Lifespan for a One-Dimensional Wave Equation with Mixed Characteristic-Time Derivative Source  (2608.28292 - Kaabi, 28 Aug 2026) in Remark 5.1 (remark labeled rem:optimality), Section 5, “The lifespan estimate and its hypotheses”