On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term
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.
Keywords
Cite
@article{arxiv.2510.17368,
title = {On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term},
author = {Yuequn Li and Alessandro Palmieri},
journal= {arXiv preprint arXiv:2510.17368},
year = {2025}
}