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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2507.07554 |
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| _version_ | 1866913935924396032 |
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| author | Guo, Lin Li, Dan |
| author_facet | Guo, Lin Li, Dan |
| contents | This paper investigates the following chemotaxis system featuring weak degradation and nonlinear motility functions
\begin{equation}\label{Model1}
\begin{cases}
u_{t} = (γ(v)u)_{xx} + r - μu, & x \in [0,L],\ t > 0,
v_{t} = v_{xx} - v + u, & x \in [0,L],\ t > 0,
\end{cases}
\end{equation} defined on the bounded interval $[0,L]$ with homogeneous Neumann boundary conditions. The motility function $γ(v)$ satisfies the regularity conditions $γ\in C^{2}[0,\infty)$ with $γ(v) > 0$ for all $v \geq 0$, and has bounded logarithmic derivative in the sense that $\sup_{v\geq 0} \frac{|γ'(v)|^{2}}{γ(v)} < \infty$. Our main results establish three fundamental properties of the system. Firstly, using energy estimate methods, we prove the existence of globally bounded solutions for all positive parameters $r, μ> 0$ and non-negative, non-trivial initial data $u_{0} \in W^{1,\infty}([0,L])$. Secondly, through the construction of an appropriate Lyapunov function, we demonstrate that all solutions $(u,v)$ converge exponentially to the unique constant equilibrium $(r/μ, r/μ)$ in the parameter regime $μ> \frac{H_{0}}{16}$, where $H_{0} := \sup_{v \geq 0} \frac{|γ'(v)|^{2}}{γ(v)}$ quantifies the maximal relative variation of the motility function. Finally, we present numerical results that not only validate the theoretical findings but also investigate the long-term behavior of solutions under diverse parameter configurations and initial conditions in two- and three-dimensional domains, providing valuable benchmarks for future research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07554 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global attractor of chemotaxis system with weak degradation and density-dependent motion Guo, Lin Li, Dan Analysis of PDEs This paper investigates the following chemotaxis system featuring weak degradation and nonlinear motility functions \begin{equation}\label{Model1} \begin{cases} u_{t} = (γ(v)u)_{xx} + r - μu, & x \in [0,L],\ t > 0, v_{t} = v_{xx} - v + u, & x \in [0,L],\ t > 0, \end{cases} \end{equation} defined on the bounded interval $[0,L]$ with homogeneous Neumann boundary conditions. The motility function $γ(v)$ satisfies the regularity conditions $γ\in C^{2}[0,\infty)$ with $γ(v) > 0$ for all $v \geq 0$, and has bounded logarithmic derivative in the sense that $\sup_{v\geq 0} \frac{|γ'(v)|^{2}}{γ(v)} < \infty$. Our main results establish three fundamental properties of the system. Firstly, using energy estimate methods, we prove the existence of globally bounded solutions for all positive parameters $r, μ> 0$ and non-negative, non-trivial initial data $u_{0} \in W^{1,\infty}([0,L])$. Secondly, through the construction of an appropriate Lyapunov function, we demonstrate that all solutions $(u,v)$ converge exponentially to the unique constant equilibrium $(r/μ, r/μ)$ in the parameter regime $μ> \frac{H_{0}}{16}$, where $H_{0} := \sup_{v \geq 0} \frac{|γ'(v)|^{2}}{γ(v)}$ quantifies the maximal relative variation of the motility function. Finally, we present numerical results that not only validate the theoretical findings but also investigate the long-term behavior of solutions under diverse parameter configurations and initial conditions in two- and three-dimensional domains, providing valuable benchmarks for future research. |
| title | Global attractor of chemotaxis system with weak degradation and density-dependent motion |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.07554 |