A remark on the blowing up of solutions to Nakao's problem (Q2124684)

In this paper, the blow-up phenomenon for a coupled system of semilinear (damped) wave equations with small data is studied, which is of the form: \(\partial_t^2 u -\Delta u +\partial_t u=|v|^p\), \(\partial_t^2 v -\Delta v =|u|^q\), for initial data on \(\mathbb{R}^n\), with \(n\ge 1\), \(p,q>1\). For a large class of small initial data (with certain nonnegative assumption), \textit{Y. Wakasugi} [in: New trends in analysis and interdisciplinary applications. Selected contributions of the 10th ISAAC congress, Macau, China, August 3--8, 2015. Basel: Birkhäuser/Springer. 545--551 (2017; Zbl 1383.35120)] has obtained certain blow up result for \(\gamma_W(n,p,q)\ge 0\). \textit{W. Chen} and \textit{M. Reissig} [J. Differ. Equations 275, 733--756 (2021; Zbl 1455.35148)] obtained partial improvement by showing blow up result (as well as an upper bound for the lifespan) for \(\gamma_{CR}(n,p,q)> 0\). The authors recover the known blow up results (except the ``critical'' case \(\gamma_W(n,p,q)=0\)), as well as proving certain upper bound for the lifespan, by employing an alternative approach of \textit{M. Ikeda} et al. [J. Differ. Equations 267, No. 9, 5165--5201 (2019; Zbl 1455.35028)]. Concerning the blow up results, the authors claim that ``all solutions blow up in finite time even for small initial data'', for which the reviewer feels inappropriate. Actually, as is clear, in all of such blow up results, we need to exploit positivity, which forces certain sign assumptions on the data.