On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term

TL;DR

This paper investigates the blow-up behavior of solutions to a coupled wave system with time-dependent damping, revealing how damping type influences the lifespan and blow-up region of solutions.

Contribution

It introduces a novel analysis of a coupled wave system with time-dependent damping, extending known results to scale-invariant and scattering cases.

Findings

Blow-up occurs in both damping scenarios under certain conditions.

The scale-invariant case shifts the blow-up region compared to classical damping.

The scattering case matches the classical Nakao problem's blow-up region.

Abstract

In this paper, we study a semilinear weakly coupled system of wave equations with power nonlinearities. More precisely, we couple (through the nonlinear terms) a wave equation and a damped wave equation with a time-dependent coefficient for the damping term. For the coefficient of the damping term we consider two cases: the scale-invariant case and the scattering producing case. By applying an iteration argument, we get a blow-up result and upper bound estimates for the lifespan of the solutions. In the scale-invariant case, we obtain a shift of the space dimension in the blow-up region for the same weakly coupled system with a classical damping (i.e. with a constant coefficient), while for the scattering producing case we find the same blow-up region as for the classical Nakao problem.