Shengquan Liu, Dongpu Li, Jiashan Zheng
Abstract:
This article concerns the chemotaxis-fluid system with double chemical signals,
$$\displaylines{
n_t + u \cdot \nabla n=\Delta n-\chi \nabla \cdot (n\nabla c)+\xi \nabla \cdot (n\nabla v) + f(n), \quad x \in \Omega, t > 0, \cr
c_t + u \cdot \nabla c=\Delta c-nc, \quad x \in \Omega,\; t > 0, \cr
v_t + u \cdot \nabla v= \Delta v-v+n, \quad x \in \Omega,\; t > 0, \cr
u_t =\Delta u+\nabla P+n\nabla \phi, \quad x \in \Omega, \;t > 0, \cr
\nabla \cdot u = 0, \quad x \in \Omega,\; t > 0,
}$$
in a smooth bounded domain $\Omega\subset \mathbb{R}^2$ with
no-flux/no-flux/no-flux/no-slip boundary conditions. Here $f(n)$ is a given
sub-logistic source function satisfying $f(s)\geq \zeta s$ for small $s\geq0$,
where $\zeta\in \mathbb{R}$, and the growth condition
$$
\liminf_{s \to \infty} \big\{-f(s) \frac{\ln s}{s^2}\big\} = \mu\in (0, \infty].
$$
We obtain the existence and uniform-in-time boundedness of classical global solutions
to the corresponding initial-boundary value problem. We assume that
the initial data and the physical coefficients satisfy
$$
\left(2 + \chi^2 + \xi^2 + 4\|c_0\|_{L^\infty(\Omega)}^2
+ 4 C_{\mathrm{p}}^2\|c_0\|_{L^\infty(\Omega)}^2
\|\nabla \phi\|_{L^\infty(\Omega)}^2 -\mu\right)^+ < \frac{1}{2 C_{\mathrm{GN}}^4 M},
$$
where \(M\) is a constant,
$$
M := \| n_0 \|_{L^1(\Omega)} + |\Omega| \inf_{\eta > 0} \frac{\sup \{ f(s) + \eta s :
s > 0 \}}{\eta},
$$
$C_{\mathrm{GN}} > 0$ is the Gagliardo-Nirenberg constant, and
$C_\mathrm{p}$ the Poincare constant. This result reveals that the sub-logistic
source plays a crucial role in preventing blow-up phenomena within the chemotaxis-fluid
system with double chemical signals.
Submitted October 21, 2025. Published February 10, 2026.
Math Subject Classifications: 35K20, 35K55, 92C17.
Key Words: Keller-Segel system; boundedness; global existence; sub-logistic source.
DOI: 10.58997/ejde.2026.10
Show me the PDF file (459 KB), TEX file for this article.
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Shengquan Liu School of Mathematics and Statistics Liaoning University Shenyang 110036, China email: shquanliu@163.com |
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Dongpu Li School of Mathematics and Statistics Liaoning University Shenyang 110036, China email: 15142132001@163.com |
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Jiashan Zheng School of Mathematics and Information Sciences Yantai University Yantai 264005, China email: zhengjiashan2008@163.com |
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