Blow-up results for a Nakao-type problem with a time-dependent damping term and derivative-type nonlinearities

In this paper, we consider a semilinear system of damped wave equations coupled through power nonlinearities of derivative-type. In particular, we consider a classical damped wave equation, i.e., with constant coefficients, and a wave equation with a time-dependent coefficient for the damping term. For this time-dependent coefficient we analyze two cases: the scale-invariant case and the scattering producing case. We prove blow-up results and derive upper bound estimates for the lifespan of local solutions. Our approach is based on an iteration argument for a couple of functionals related to the components of a local solution.