Refined criteria toward boundedness in an attraction–repulsion chemotaxis system with nonlinear productions

Abstract

We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant, whose formulation can be schematized as (⋄) $\{\begin{array}{ll}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v)+\xi \mathrm{\nabla }\cdot (u\mathrm{\nabla }w)& \mathrm{in}\mathrm{\Omega }\times (0,T_{max}),\\ \tau v_t=\mathrm{\Delta }v-\phi (t,v)+f(u)& \mathrm{in}\mathrm{\Omega }\times (0,T_{max}),\\ \tau w=\mathrm{\Delta }w-\psi (t,w)+g(u)& \mathrm{in}\mathrm{\Omega }\times (0,T_{max}).\end{array}$(⋄) In this problem, Ω is a bounded and smooth domain of ${\mathbb{R}}^n$, for $n\ge 2$, $\chi ,\xi >0$, $f(u)$, $g(u)$ reasonably regular functions generalizing, respectively, the prototypes $f(u)=\alpha u^k$ and $g(u)=\gamma u^l$, for some $k,l,\alpha ,\gamma >0$ and all $u\ge 0$. Moreover, $\phi (t,v)$ and $\psi (t,w)$ have specific expressions, $\tau \in \{0,1\}$ and $\mathrm{\Theta }:=\chi \alpha -\xi \gamma$. Once for any sufficiently smooth $u(x,0)=u_0(x)\ge 0$, $\tau v(x,0)=\tau v_0(x)\ge 0$ and $\tau w(x,0)=\tau w_0(x)\ge 0,$ the local well-posedness of problem ($◊$) is ensured, and we establish for the classical solution $(u,v,w)$ defined in $\mathrm{\Omega }\times (0,T_{max})$ that the life span is indeed $T_{max}=\mathrm{\infty }$ and u, v and w are uniformly bounded in $\mathrm{\Omega }\times (0,\mathrm{\infty })$ in the following cases:

1. For $\phi (t,v)=\beta v$, $\beta >0$, $\psi (t,w)=\delta w$, $\delta >0$ and $\tau =0$, provided

   	(I.1)	k<l;
   	(I.2)	$k,l\in (0,\frac{2}{n})$;
   	(I.3)	k = l and $\mathrm{\Theta }<0$, or $l=k\in (0,\frac{2}{n})$ and $\mathrm{\Theta }\ge 0$.

2. For $\phi (t,v)=\beta v$, $\beta >0$, $\psi (t,w)=\delta w$, $\delta >0$ and $\tau =1$, whenever

   	(II.1)	$l,k\in (0,\frac{1}{n}]$;
   	(II.2)	$l\in (\frac{1}{n},\frac{1}{n}+\frac{2}{n^2+4})$ and $k\in (0,\frac{1}{n}]$, or $k\in (\frac{1}{n},\frac{1}{n}+\frac{2}{n^2+4})$ and $l\in (0,\frac{1}{n}]$;
   	(II.3)	$l,k\in (\frac{1}{n},\frac{1}{n}+\frac{2}{n^2+4})$.

3. For $\phi (t,v)=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}f(u)$ and $\psi (t,w)=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}g(u)$ and $\tau =0$, under the assumptions k<l or (I.3)).

In particular, in this paper we partially improve what derived in Viglialoro [Influence of nonlinear production on the global solvability of an attraction-repulsion chemotaxis system. Math Nachr. 2021;294(12):2441–2454] and solve an open question given in Liu and Li [Finite-time blowup in attraction-repulsion systems with nonlinear signal production. Nonlinear Anal Real World Appl. 2021;61:Paper No. 103305, 21]. Finally, the research is complemented with numerical simulations in bi-dimensional domains.

Keywords:

• Chemotaxis
• global existence
• boundedness
• nonlinear production

Mathematics Subject Classification 2010:

• Primary: 35A01
• 35K55
• 35Q92. Secondary:92C17

Acknowledgments

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) ofthe Istituto Nazionale di Alta Matematica (INdAM).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors are partially supported by the research projects Evolutive and Stationary Partial Differential Equations with a Focus on Biomathematics (2019, Grant Number: F72F20000200007), Analysis of PDEs in connection with real phenomena (2021, Grant Number: F73C22001130007), funded by https://www.fondazionedisardegna.it/, Fondazione di Sardegna. GV is also supported by MIUR (Italian Ministry of Education, University and Research) Prin 2017 Nonlinear Differential Problems via Variational, Topological and Set-valued Methods (Grant Number: 2017AYM8XW).