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Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth

Viglialoro, Giuseppe and Woolley, Thomas 2017. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete and Continuous Dynamical Systems - Series B
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Abstract

In this paper we study the chemotaxis-system \begin{equation*} \begin{cases} u_{t}=\Delta u-\chi \nabla \cdot (u\nabla v)+g(u) & x\in \Omega, t>0, \\ v_{t}=\Delta v-v+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: Completion
Status: In Press
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: American Institute of Mathematical Sciences
ISSN: 1531-3492
Last Modified: 20 Jul 2017 00:05
URI: http://orca.cf.ac.uk/id/eprint/102525

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