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This work considers the chemotaxis model with logistic growth and indirect signal production {ut=∇⋅(D(u)∇u)−∇⋅(u∇v)+f(u),x∈Ω,t>0,vt=Δv+w−v,x∈Ω,t>0,wt=Δw+u−w,x∈Ω,t>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \textstyle\begin{cases} \displaystyle u_{t}= \nabla \cdot (D(u) \nabla u)- \nabla \cdot (u \nabla v)+f(u) , & \;\; x\in \Omega ,\; t>0, \\ \displaystyle v_{t}=\Delta v + w- v, & \;\; x\in \Omega ,\; t>0, \\ \displaystyle w_{t}=\Delta w+ u-w, & \;\; x\in \Omega ,\; t>0 \end{cases} $$\end{document} under homogeneous Neumann boundary condition in a smooth bounded domain Ω∈Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\Omega \in {\mathbb{R}}^{n}$\end{document} (n≥3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n\ge 3$\end{document}). In this model, D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$D$\end{document} and f\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$f$\end{document} are smooth functions satisfying D(u)≥D1uγ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$D(u)\ge D_{1}u^{\gamma }$\end{document} and f(u)≤μ(u−uα)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$f(u)\le \mu (u-u^{\alpha })$\end{document} with D1>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$D_{1}>0$\end{document}, γ∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\gamma \in {\mathbb{R}}$\end{document}, α≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha \ge 2$\end{document} and μ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mu >0$\end{document}. Then if the assumption αn+γ2>12\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\frac{\alpha }{n}+\frac{\gamma }{2}>\frac{1}{2}$\end{document} holds, the system possesses a classical solution which is global in time and bounded.
Acta Applicandae Mathematicae – Springer Journals
Published: Dec 1, 2021
Keywords: Chemotaxis; Indirect signal production; Logistic growth; Quasilinear; Global boundedness
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