A piezoelectric beam model with nonlinear dampings and supercritical sources
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918374582255616 |
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| author | Liao, Menglan Feng, Baowei |
| author_facet | Liao, Menglan Feng, Baowei |
| contents | This paper aims to investigate a three-dimensional fully magnetic effected piezoelectric beam model with strong sources and nonlinear interior dampings. By employing nonlinear semigroups and the theory of monotone operators, the existence of local weak solutions is established. By the potential well method, we obtain the global existence of potential well solutions. Decay rates of the total energy are obtained in terms of the behavior of the damping terms. The main advantage in this work is that the stabilization estimate does not generate lower-order terms, and in addition we remove some strong conditions in previous results to obtain a weaker energy decay. Finally, when the initial total energy is negative, positive but small, respectively, the blow-up results for weak solutions if the source terms are stronger than damping terms are obtained according to the differential inequality technique. Moreover, if interior dampings are linear, a blow-up result with arbitrarily high initial energy is established by the concavity method and an upper bound for the blow-up time is also derived. All results are independent of any relation among the model coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04049 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A piezoelectric beam model with nonlinear dampings and supercritical sources Liao, Menglan Feng, Baowei Analysis of PDEs 35L51, 35L53, 35B35, 35B44 This paper aims to investigate a three-dimensional fully magnetic effected piezoelectric beam model with strong sources and nonlinear interior dampings. By employing nonlinear semigroups and the theory of monotone operators, the existence of local weak solutions is established. By the potential well method, we obtain the global existence of potential well solutions. Decay rates of the total energy are obtained in terms of the behavior of the damping terms. The main advantage in this work is that the stabilization estimate does not generate lower-order terms, and in addition we remove some strong conditions in previous results to obtain a weaker energy decay. Finally, when the initial total energy is negative, positive but small, respectively, the blow-up results for weak solutions if the source terms are stronger than damping terms are obtained according to the differential inequality technique. Moreover, if interior dampings are linear, a blow-up result with arbitrarily high initial energy is established by the concavity method and an upper bound for the blow-up time is also derived. All results are independent of any relation among the model coefficients. |
| title | A piezoelectric beam model with nonlinear dampings and supercritical sources |
| topic | Analysis of PDEs 35L51, 35L53, 35B35, 35B44 |
| url | https://arxiv.org/abs/2505.04049 |