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

  • * Corresponding author: Giuseppe Viglialoro

    * Corresponding author: Giuseppe Viglialoro 

TEW would like to thank St John's College, Oxford and the Mathematical Biosciences Institute (MBI) at Ohio State University, for financially supporting this research through the National Science Foundation grant DMS 1440386 and BBSRC grant BKNXBKOO BK00.16

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  • In this paper we study the chemotaxis-system

    $\begin{equation*}\begin{cases}u_{t}=Δ u-χ \nabla · (u\nabla v)+g(u)&x∈ Ω, t>0, \\v_{t}=Δ v-v+u&x∈ Ω, t>0,\end{cases}\end{equation*}$

    defined in a convex smooth and bounded domain $Ω$ of $\mathbb{R}^n$, $n≥ 1$, with $χ>0$ and endowed with homogeneous Neumann boundary conditions. The source $g$ behaves similarly to the logistic function and satisfies $g(s)≤ a -bs^α$, for $s≥ 0$, with $a≥ 0$, $b>0$ and $α>1$. Continuing the research initiated in [33], where for appropriate $1 < p < α < 2$ and $(u_0,v_0) ∈ C^0(\bar{Ω})× C^2(\bar{Ω})$ the global existence of very weak solutions $(u,v)$ to the system (for any $n≥ 1$) is shown, we principally study boundedness and regularity of these solutions after some time. More precisely, when $n=3$, we establish that

    -for all $τ>0$ an upper bound for $\frac{a}{b}, ||u_0||_{L^1(Ω)}, ||v_0||_{W^{2,α}(Ω)}$ can be prescribed in a such a way that $(u,v)$ is bounded and Hölder continuous beyond $τ$;

    -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 Hölder continuous beyond $T$.

    Finally, we illustrate the range of dynamics present within the chemotaxis system in one, two and three dimensions by means of numerical simulations.

    Mathematics Subject Classification: Primary: 35B40, 35K55, 35B65, 92C17; Secondary: 35Q92.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Simulations of system (45) in one dimension with varying value of $\alpha$, given beneath each subfigure. Each subfigure contains the system evaluated at the time points $t=1$, 10, 50 and 100. The remaining parameters values are $a=1$, $b=1.1$ and $\chi=6$. The domain was discretised into 1000 equally spaced points

    Figure 2.  Simulations of system (45) in one dimension. The simulations are nearly identical to those seen in Figure 1(a). However, each simulation involves a single parameter change. Specifically, in (a) a larger initial condition for $u$ was used (100 was added to the mean); in (b) the parameter $b$ was reduced to 0.2; Finally, in (c) the spatial solution domain has been reduced from 10 to 1

    Figure 3.  Simulations of system (45) in two dimensions with varying value of $\alpha$, given beneath each subfigure. Evolution time shown above each subfigure. The remaining parameters values are $a=1$, $b=1.1$ and $\chi=6$. The domain was triangulated into 24, 968 finite elements. The figure inset of (b) shows the full extent of the peak, which is growing without bound

    Figure 4.  Simulations of system (45) illustrating the density of $u$ in three dimensions with varying value of $\alpha$, given beneath each subfigure. Evolution time shown above each subfigure. The remaining parameters values are $a=1$, $b=1.1$ and $\chi=6$. The domain was discretised into 1, 139, 254 voxel elements. Apart from the light grey ball illustrating the boundary of the solution domain the images illustrate isosurfaces of the solution (i.e. surface that represent points of a constant value, thus, they are the three-dimensional analogue of contours). In Figure (a) there are five isosurfaces of value 1, 1.25, 1.5 1.75 and 2, coloured, yellow, green, blue, red and black, respectively. In Figure (b) there are three isosurfaces of value 1, 10, and $10^6$, coloured, yellow, blue and black, respectively

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