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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.02583 |
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| _version_ | 1866909940232224768 |
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| author | Tao, Qiang Wang, Dehua Yang, Ying Zhong, Meifang |
| author_facet | Tao, Qiang Wang, Dehua Yang, Ying Zhong, Meifang |
| contents | In this paper, we investigate the optimal large-time behavior of the global solution to a singular chemotaxis system in the whole space $\mathbb{R}^d$ with $d=2,3$. Assuming that the initial data is sufficiently close to an equilibrium state, we first prove the $k$-th order spatial derivative of the global solution converges to its corresponding equilibrium at the optimal rate $(1+t)^{-(\frac{d}{4}+\frac{k}{2})}$, which improve upon the result in [37]. Then, for well-chosen initial data, we also establish lower bounds on the convergence rates, which match those of the heat equation. Our proof relies on a Cole-Hopf type transformation, delicate spectral analysis, the Fourier splitting technique, and energy methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of solutions to a singular chemotaxis system in multi-dimensions Tao, Qiang Wang, Dehua Yang, Ying Zhong, Meifang Analysis of PDEs In this paper, we investigate the optimal large-time behavior of the global solution to a singular chemotaxis system in the whole space $\mathbb{R}^d$ with $d=2,3$. Assuming that the initial data is sufficiently close to an equilibrium state, we first prove the $k$-th order spatial derivative of the global solution converges to its corresponding equilibrium at the optimal rate $(1+t)^{-(\frac{d}{4}+\frac{k}{2})}$, which improve upon the result in [37]. Then, for well-chosen initial data, we also establish lower bounds on the convergence rates, which match those of the heat equation. Our proof relies on a Cole-Hopf type transformation, delicate spectral analysis, the Fourier splitting technique, and energy methods. |
| title | Asymptotic behavior of solutions to a singular chemotaxis system in multi-dimensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.02583 |