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 - <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>