Large time behavior of solution to a parabolic-elliptic chemotaxis system with weak singular sensitivity and logistic kinetics: Boundedness, persistence, stability
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866908636999057408 |
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| author | Kurt, Halil ibrahim |
| author_facet | Kurt, Halil ibrahim |
| contents | This paper deals with the long-term behavior of positive solutions for the following parabolic-elliptic chemotaxis competition system with weak singular sensitivity and logistic source
\begin{equation}
\label{abstract-eq} \begin{cases} u_t=Δu-χ\nabla\cdot (\frac{u}{v^λ} \nabla v) +ru- μu^2, \quad &x\in Ω,\cr 0=Δv- αv +βu,\quad &x\in Ω, \cr \frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0,\quad &x\in\partialΩ, \end{cases}\, \end{equation} where $Ω\subset \mathbb{R}^N (N \ge 2)$ is a smooth bounded domain, the parameters $χ,\, r, \, μ, \, α,\,β$ are positive constants and $λ\in (0,1).$
In this article, for all suitably smooth initial data $u_0\in C^0(\barΩ)$ with $u_0 \not \equiv 0,$ it has been proven that:
First, there exists $μ> μ_1^*(p,λ,χ,β)$ such that any globally defined positive solution is $L^p(Ω)$-bounded with $p \ge 2.$ Next, there exists $μ> μ_2^*(N,λ,χ,β)$ such that any globally defined classical solutions is globally bounded. Third, there exists $μ> μ_3^*(N,λ,χ,β)$ such that any globally defined positive solution is uniformly bounded above and below eventually by some positive constants that are independent of its initial function $u_0.$ Last, there exists $μ> μ_4^*(N,λ,χ,α,β,r,Ω)$ such that any globally bounded classical solution to system (0.1) exponentially converges to the constant steady state $(\frac{r}μ,\fracβα\frac{r}μ).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15852 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large time behavior of solution to a parabolic-elliptic chemotaxis system with weak singular sensitivity and logistic kinetics: Boundedness, persistence, stability Kurt, Halil ibrahim Analysis of PDEs Dynamical Systems This paper deals with the long-term behavior of positive solutions for the following parabolic-elliptic chemotaxis competition system with weak singular sensitivity and logistic source \begin{equation} \label{abstract-eq} \begin{cases} u_t=Δu-χ\nabla\cdot (\frac{u}{v^λ} \nabla v) +ru- μu^2, \quad &x\in Ω,\cr 0=Δv- αv +βu,\quad &x\in Ω, \cr \frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0,\quad &x\in\partialΩ, \end{cases}\, \end{equation} where $Ω\subset \mathbb{R}^N (N \ge 2)$ is a smooth bounded domain, the parameters $χ,\, r, \, μ, \, α,\,β$ are positive constants and $λ\in (0,1).$ In this article, for all suitably smooth initial data $u_0\in C^0(\barΩ)$ with $u_0 \not \equiv 0,$ it has been proven that: First, there exists $μ> μ_1^*(p,λ,χ,β)$ such that any globally defined positive solution is $L^p(Ω)$-bounded with $p \ge 2.$ Next, there exists $μ> μ_2^*(N,λ,χ,β)$ such that any globally defined classical solutions is globally bounded. Third, there exists $μ> μ_3^*(N,λ,χ,β)$ such that any globally defined positive solution is uniformly bounded above and below eventually by some positive constants that are independent of its initial function $u_0.$ Last, there exists $μ> μ_4^*(N,λ,χ,α,β,r,Ω)$ such that any globally bounded classical solution to system (0.1) exponentially converges to the constant steady state $(\frac{r}μ,\fracβα\frac{r}μ).$ |
| title | Large time behavior of solution to a parabolic-elliptic chemotaxis system with weak singular sensitivity and logistic kinetics: Boundedness, persistence, stability |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2411.15852 |