On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911221089828864 |
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| author | Li, Yuequn Palmieri, Alessandro |
| author_facet | Li, Yuequn Palmieri, Alessandro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17368 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term Li, Yuequn Palmieri, Alessandro Analysis of PDEs 35B44, 35G55, 35L52, 35L71 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. |
| title | On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term |
| topic | Analysis of PDEs 35B44, 35G55, 35L52, 35L71 |
| url | https://arxiv.org/abs/2510.17368 |