Blow-up of solutions to Nakao's problem via an iteration argument

In this paper, we consider blow-up behavior of weak solutions to a weakly coupled system for a semilinear damped wave equation and a semilinear wave equation in ${\mathbb{R}}^n$. This problem is part of the so-called Nakao’s problem proposed by Professor Mitsuhiro Nakao (Kyushu University) for a critical relation between the exponents $p$ and $q$. By applying an iteration method for unbounded multipliers with a slicing procedure, we prove blow-up of weak solutions for Nakao’s problem even for small data. We improve the blow-up result and upper bound estimates for lifespan comparing with the previous research, especially, in higher dimensional cases.