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

This paper is devoted to studying blow-up of solutions to some system of a semilinear damped wave equation and a semilinear wave equation for small data. In particular, we prove that if the exponents of the nonlinear terms satisfy suitable conditions, all solutions blow up in finite time even for small initial data, and we are also concerned with the upper estimate of lifespan of blowing-up solutions. A study on blowing up of solutions for this problem was originated by Wakasugi (2017), which used the test function method, and was subsequently partially improved by Chen-Reissig (2021) with an iteration argument. However, there is a gap between these two results. Our main aim in this paper is to bridge this gap while also providing a more concise proof.