Determine the sharp asymptotic lifespan constant

Determine whether the limit \(\lim_{\eta\to0^+}\eta^{m+n-1}T(\eta)\) exists for the one-dimensional semilinear wave equation \(v_{tt}-v_{xx}=|v_t+v_x|^m|v_t|^n\), and, if it exists, identify it in terms of the initial profiles \(\phi\) and \(\psi\).

Background

The paper establishes two-sided lifespan bounds of order T(η)≍η−(m+n−1)T(\eta)\asymp\eta^{-(m+n-1)} for compactly supported nontrivial data satisfying the one-sided characteristic condition ψ−ϕ′≥0\psi-\phi'\geq0. The lower- and upper-bound arguments produce explicit constants, but these constants do not generally match; the authors note that the resulting ratio is large. The unresolved issue is whether the rescaled lifespan has a limiting constant as the amplitude η\eta tends to zero, and whether that constant can be expressed directly through the initial data profiles.

References

Three questions are left open. The first is the determination of the sharp multiplicative constants, that is, whether $\varrho{\vartheta(p,q)}T{*}(\varrho)$ converges as $\varrho\to\infty$ and, if so, the identification of its limit in terms of the profiles $f$ and $g$; the constants produced here and in are far apart, and neither argument is designed to be sharp.

— Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping  (2609.10819 - Kaabi, 9 Sep 2026) in Section 5, Concluding remarks and open problems

What is not determined here is the constant: the ratio C/c produced by the two arguments is large, and identifying \lim_{\eta\to0+}\eta{m+n-1}T(\eta), if it exists, in terms of the profiles \phi and \psi remains open.

— 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”

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”