Asymptotically almost Periodic Solutions of the parabolic-parabolic Keller-Segel systems on bounded domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912423378681856 |
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| author | Xuan, Pham Truong |
| author_facet | Xuan, Pham Truong |
| contents | In this paper, we investigate the existence, uniqueness, and exponential decay of asymptotically almost periodic (AAP-) mild solutions for the parabolic-parabolic Keller-Segel systems on a bounded domain $Ω\subset \mathbb{R}^n$ with a smooth boundary. First, we establish the well-posedness of mild solutions for the corresponding linear systems by utilizing the dispersive and smoothing estimates of the Neumann heat semigroup on the bounded domain $Ω$. We then prove the existence and uniqueness of AAP-mild solutions for the linear systems by providing a Massera-type principle. Next, using results of the linear systems and fixed-point arguments, we derive the well-posedness of such solutions for the Keller-Segel systems. Finally, the exponential decay of these solutions is demonstrated through a Gronwall-type inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_00347 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotically almost Periodic Solutions of the parabolic-parabolic Keller-Segel systems on bounded domains Xuan, Pham Truong Analysis of PDEs 35A01, 35B65, 35Q30, 43A60, 92C17 In this paper, we investigate the existence, uniqueness, and exponential decay of asymptotically almost periodic (AAP-) mild solutions for the parabolic-parabolic Keller-Segel systems on a bounded domain $Ω\subset \mathbb{R}^n$ with a smooth boundary. First, we establish the well-posedness of mild solutions for the corresponding linear systems by utilizing the dispersive and smoothing estimates of the Neumann heat semigroup on the bounded domain $Ω$. We then prove the existence and uniqueness of AAP-mild solutions for the linear systems by providing a Massera-type principle. Next, using results of the linear systems and fixed-point arguments, we derive the well-posedness of such solutions for the Keller-Segel systems. Finally, the exponential decay of these solutions is demonstrated through a Gronwall-type inequality. |
| title | Asymptotically almost Periodic Solutions of the parabolic-parabolic Keller-Segel systems on bounded domains |
| topic | Analysis of PDEs 35A01, 35B65, 35Q30, 43A60, 92C17 |
| url | https://arxiv.org/abs/2412.00347 |