Volume 1, Issue 4
Boundedness and Asymptotic Stability in a Chemotaxis Model with Signal-Dependent Motility and Nonlinear Signal Secretion

Commun. Math. Anal. Appl., 1 (2022), pp. 568-589.

Published online: 2022-10

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• Abstract

In the present study, we consider the following parabolic-elliptic chemotaxis system: $$\begin{cases} u_t=∇·(γ(v)∇u−u\chi(v)∇v) +λu−\mu u^σ , x∈Ω, \ t>0, \\ 0=∆v−v+u^κ, x∈Ω, \ t>0,\end{cases}$$ where $Ω⊂\mathbb{R}^n(n≥2)$ is a smooth and bounded domain, $λ>0,$ $\mu>0,$ $σ>1,$ $κ>0.$ Under appropriate assumptions on $γ(v)$ and $\chi(v),$ we obtain the global boundedness of solutions when $κn<2$ or $κn ≥2,$ $σ ≥κ+1,$ which generalize the previous result to the case with nonlinear signal secretion and superlinear logistic term when $n≥2.$ Moreover, if adding additional conditions $σ≥2κ$ and $\mu$ is sufficiently large, it is shown that the global solution $(u,v)$ converges to $$\left(\left(\frac{λ}{\mu}\right)^{\frac{1}{σ−1}}, \left(\frac{λ}{\mu}\right)^{\frac{κ}{σ−1}}\right)$$ exponentially as $t→∞.$

• AMS Subject Headings

35K35, 35Q30, 35B40

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COPYRIGHT: © Global Science Press

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@Article{CMAA-1-568, author = {Tu , XinyuMu , ChunlaiQiu , Shuyan and Zhang , Jing}, title = {Boundedness and Asymptotic Stability in a Chemotaxis Model with Signal-Dependent Motility and Nonlinear Signal Secretion}, journal = {Communications in Mathematical Analysis and Applications}, year = {2022}, volume = {1}, number = {4}, pages = {568--589}, abstract = {

In the present study, we consider the following parabolic-elliptic chemotaxis system: $$\begin{cases} u_t=∇·(γ(v)∇u−u\chi(v)∇v) +λu−\mu u^σ , x∈Ω, \ t>0, \\ 0=∆v−v+u^κ, x∈Ω, \ t>0,\end{cases}$$ where $Ω⊂\mathbb{R}^n(n≥2)$ is a smooth and bounded domain, $λ>0,$ $\mu>0,$ $σ>1,$ $κ>0.$ Under appropriate assumptions on $γ(v)$ and $\chi(v),$ we obtain the global boundedness of solutions when $κn<2$ or $κn ≥2,$ $σ ≥κ+1,$ which generalize the previous result to the case with nonlinear signal secretion and superlinear logistic term when $n≥2.$ Moreover, if adding additional conditions $σ≥2κ$ and $\mu$ is sufficiently large, it is shown that the global solution $(u,v)$ converges to $$\left(\left(\frac{λ}{\mu}\right)^{\frac{1}{σ−1}}, \left(\frac{λ}{\mu}\right)^{\frac{κ}{σ−1}}\right)$$ exponentially as $t→∞.$

}, issn = {2790-1939}, doi = {https://doi.org/10.4208/cmaa.2022-0018}, url = {http://global-sci.org/intro/article_detail/cmaa/21122.html} }
TY - JOUR T1 - Boundedness and Asymptotic Stability in a Chemotaxis Model with Signal-Dependent Motility and Nonlinear Signal Secretion AU - Tu , Xinyu AU - Mu , Chunlai AU - Qiu , Shuyan AU - Zhang , Jing JO - Communications in Mathematical Analysis and Applications VL - 4 SP - 568 EP - 589 PY - 2022 DA - 2022/10 SN - 1 DO - http://doi.org/10.4208/cmaa.2022-0018 UR - https://global-sci.org/intro/article_detail/cmaa/21122.html KW - Chemotaxis, boundedness, large time behavior, signal-dependent motility, nonlinear signal secretion. AB -

In the present study, we consider the following parabolic-elliptic chemotaxis system: $$\begin{cases} u_t=∇·(γ(v)∇u−u\chi(v)∇v) +λu−\mu u^σ , x∈Ω, \ t>0, \\ 0=∆v−v+u^κ, x∈Ω, \ t>0,\end{cases}$$ where $Ω⊂\mathbb{R}^n(n≥2)$ is a smooth and bounded domain, $λ>0,$ $\mu>0,$ $σ>1,$ $κ>0.$ Under appropriate assumptions on $γ(v)$ and $\chi(v),$ we obtain the global boundedness of solutions when $κn<2$ or $κn ≥2,$ $σ ≥κ+1,$ which generalize the previous result to the case with nonlinear signal secretion and superlinear logistic term when $n≥2.$ Moreover, if adding additional conditions $σ≥2κ$ and $\mu$ is sufficiently large, it is shown that the global solution $(u,v)$ converges to $$\left(\left(\frac{λ}{\mu}\right)^{\frac{1}{σ−1}}, \left(\frac{λ}{\mu}\right)^{\frac{κ}{σ−1}}\right)$$ exponentially as $t→∞.$

Xinyu Tu, Chunlai Mu, Shuyan Qiu & Jing Zhang. (2022). Boundedness and Asymptotic Stability in a Chemotaxis Model with Signal-Dependent Motility and Nonlinear Signal Secretion. Communications in Mathematical Analysis and Applications. 1 (4). 568-589. doi:10.4208/cmaa.2022-0018
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