Influence of nonlinear production on the global solvability of an attraction‐repulsion chemotaxis system

Giuseppe Viglialoro

Mathematische Nachrichten, 2021, vol. 294, issue 12, 2441-2454

Abstract: This paper is dedicated to the attraction‐repulsion chemotaxis‐system ⋄$\Diamond$ ut=Δu−χ∇·(u∇v)+ξ∇·(u∇w)inΩ×(0,Tmax),0=Δv+f(u)−βvinΩ×(0,Tmax),0=Δw+g(u)−δwinΩ×(0,Tmax),\begin{equation} \hspace*{65pt}\left\{ \def\eqcellsep{&}\begin{array}{ll} u_{ t}=\Delta u -\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w) & \text{in }\Omega \times (0,T_{\mathrm{max}}), \\ 0=\Delta v+f(u)-\beta v & \text{in } \Omega \times (0,T_{\mathrm{max}}), \\ 0=\Delta w+g(u)-\delta w & \text{in } \Omega \times (0,T_{\mathrm{max}}), \end{array} \right. \end{equation}defined in Ω, a smooth and bounded domain of Rn$\mathbb {R}^n$, with n≥2$n\ge 2$. Moreover, β,δ,χ,ξ>0$\beta ,\delta ,\chi ,\xi >0$ and f,g$f, g$ are suitably regular functions generalizing, for u≥0$u\ge 0$ and α, γ>0$\gamma >0$ the prototypes f(u)=αus$f(u)=\alpha u^s$, s>0$s>0$, and g(u)=γur$g(u)=\gamma u^r$, r≥1$r\ge 1$. We focus our analysis on the value Tmax∈(0,∞]$T_{\mathrm{max}}\in (0,\infty ]$, establishing the temporal interval of existence of solutions (u,v,w)$(u,v,w)$ to problem (⋄$\Diamond$). When zero‐flux boundary conditions are fixed, we prove the following results, all excluding chemotactic collapse scenarios under certain correlations between the attraction and repulsive effects describing the model. To be precise, for every α,β,γ,δ,χ>0$\alpha ,\beta ,\gamma ,\delta ,\chi >0$, and r>s≥1$r>s\ge 1$ (resp. s>r≥1$s>r\ge 1$), there exists ξ∗>0$\xi ^*>0$ (resp. ξ∗>0$\xi _*>0$) such that if ξ>ξ∗$\xi >\xi ^*$ (resp. ξ≥ξ∗$\xi \ge \xi _*$), any sufficiently regular initial datum u0(x)≥0$u_0(x)\ge 0$ (resp. u0(x)≥0$u_0(x)\ge 0$ enjoying some smallness assumptions) produces a unique classical solution (u,v,w)$(u,v,w)$ to problem (⋄$\Diamond$) which is global, i.e. Tmax=∞$T_{\mathrm{max}}=\infty$, and such that u, v and w are uniformly bounded. Conversely, the same conclusion holds true for every α,β,γ,δ,χ,ξ>0$\alpha ,\beta ,\gamma ,\delta ,\chi ,\xi >0$, 0

Date: 2021
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.201900465

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:294:y:2021:i:12:p:2441-2454

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().