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arXiv · 2510.17368

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

Yuequn Li·Alessandro Palmieri
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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.

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Yuequn Li, Alessandro Palmieri. 2025-10-20. On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term. https://arxiv.org/abs/2510.17368

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