Viglialoro, Giuseppe and Woolley, Thomas 2018. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete and Continuous Dynamical Systems  Series B 23 (8) , pp. 30233045. 10.3934/dcdsb.2017199 

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Abstract
In this paper we study the chemotaxissystem \begin{equation*} \begin{cases} u_{t}=\Delta u\chi \nabla \cdot (u\nabla v)+g(u) & x\in \Omega, t>0, \\ v_{t}=\Delta vv+u & x\in \Omega, t>0, \end{cases} \end{equation*} defined in a convex smooth and bounded domain $\Omega$ of $\R^n$, $n\geq 1$, with $\chi>0$ and endowed with homogeneous Neumann boundary conditions. The source $g$ behaves similarly to the logistic function and satisfies $g(s)\leq a bs^\alpha$, for $s\geq 0$, with $a\geq 0$, $b>0$ and $\alpha>1$. Continuing the research initiated in \citep{ViglialoroVeryWeak}, where for appropriate $1<p<\alpha<2$ and $(u_0,v_0) \in C^0(\bar{\Omega})\times C^2(\bar{\Omega})$ the global existence of very weak solutions $(u,v)$ to the system (for any $n\geq 1$) is shown, we principally study boundedness and regularity of these solutions after some time. More precisely, when $n=3$, we establish that \begin{enumerate} \item [] for all $\tau>0$ an upper bound for $\frac{a}{b}, u_0_{L^1(\Omega)}, v_0_{W^{2,\alpha}(\Omega)}$ can be prescribed in a such a way that $(u,v)$ is bounded and H\"{o}lder continuous beyond $\tau$; \item [] for all $(u_0,v_0)$, and sufficiently small ratio $\frac{a}{b}$, there exists a $T>0$ such that $(u,v)$ is bounded and H\"{o}lder continuous beyond $T$. \end{enumerate} Finally, we illustrate the range of dynamics present within the chemotaxis system in one, two and three dimensions by means of numerical simulations.
Item Type:  Article 

Date Type:  Publication 
Status:  Published 
Schools:  Mathematics 
Subjects:  Q Science > QA Mathematics 
Publisher:  American Institute of Mathematical Sciences 
ISSN:  15313492 
Date of First Compliant Deposit:  19 July 2017 
Date of Acceptance:  26 June 2017 
Last Modified:  01 Oct 2019 02:19 
URI:  http://orca.cf.ac.uk/id/eprint/102525 
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