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

Viglialoro, Giuseppe and Woolley, Thomas E. ORCID: https://orcid.org/0000-0001-6225-5365 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. 3023-3045. 10.3934/dcdsb.2017199

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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 $10$ 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:	1531-3492
Date of First Compliant Deposit:	19 July 2017
Date of Acceptance:	26 June 2017
Last Modified:	22 Nov 2024 08:15
URI:	https://orca.cardiff.ac.uk/id/eprint/102525

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