@Article{CMAA-1-4,
author = {Xinyu, Tu and Mu, Chunlai and Shuyan, Qiu 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 = {<p style="text-align: justify;">In the present study, we consider the following parabolic-elliptic
chemotaxis system: $$\begin{cases} u_t=∇·(γ(v)∇u−u\chi(v)∇v) +λu−\mu u^σ
, x∈Ω, \ t&gt;0, \\ 0=∆v−v+u^κ, x∈Ω, \ t&gt;0,\end{cases}$$ where $Ω⊂\mathbb{R}^n(n≥2)$ is a smooth and bounded domain, $λ&gt;0,$ $\mu&gt;0,$ $σ&gt;1,$ $κ&gt;0.$ Under appropriate assumptions on $γ(v)$ and $\chi(v),$ we obtain the global boundedness of solutions when $κn&lt;2$ 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 $t→∞.$</p>},
issn = {2790-1939},
doi = {https://doi.org/10.4208/cmaa.2022-0018},
url = {https://global-sci.com/article/81441/boundedness-and-asymptotic-stability-in-a-chemotaxis-model-with-signal-dependent-motility-and-nonlinear-signal-secretion}
}