Blowup for a Damped Wave Equation with Mass and General Nonlinear Memory

Abstract

We investigate the blowup conditions to the Cauchy problem for a semilinear wave equation with scale-invariant damping, mass and general nonlinear memory term (see Eq. (1.1) in the Introduction). We first establish a local (in time) existence result for this problem by Banach’s fixed point theorem, where Palmieri’s decay estimates on the solution to the corresponding linear homogeneous equation play an essential role in the proof. We then formulate a blowup result for energy solutions by applying the iteration argument together with the test function method.