Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 26, pp. 1-20.

Title: Global well-posedness to a multidimensional parabolic-elliptic-elliptic attraction-repulsion chemotaxis system

Author: Ling Liu (Jilin Jianzhu Univ., Changchun, China)
  
Abstract:
In this article we study the initial-boundary value problem for the 
attraction-repulsion chemotaxis system
$$
\displaylines{
u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w),
\quad x\in\Omega,; t>0,\cr
0=\Delta v-\beta v+\alpha u, \quad x\in\Omega,\; t>0,\cr
0=\Delta w-\delta w+\gamma u, \quad x\in\Omega,\; t>0,\cr
\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}
=\frac{\partial w}{\partial \nu}=0, \quad x\in\partial\Omega,\; t>0,\cr
u(x,0)=u_0(x), \quad x\in\Omega,
}$$
with homogenous Neumann boundary conditions in a multidimensional 
bounded domain $\Omega\subset\mathbb{R}^N$ $(1\leq N\leq 4)$
with smooth boundary, where $\chi$, $\xi$, $\alpha$, $\beta$, $\delta$ 
and $\gamma$ are positive constants. 
We prove that under the assumption  $\chi\alpha=\xi\gamma$ the
corresponding system possesses a unique global bounded classical solution 
in the cases $N\leq 3$ or 
$ \lambda_0 \gamma\delta\xi \| u_0\|^{10/7}_{L^1(\Omega)}
< \frac{1}{C_{GN}}$ and $N=4$. Moreover, the large
time behavior of solutions is also investigated. 
Specially, when $\chi\alpha=\xi\gamma$, the  solution of the 
system  converges to
$(\bar{u}_0,\frac\alpha\beta\bar{u}_0,\frac\gamma\delta\bar{u}_0)$
exponentially if $\|u_0\|_{L^\infty(\Omega)}$ is small. 


Submitted January 5, 2025. Published March 11, 2025
Math Subject Classifications: 35K55, 35Q92, 35Q35, 92C17
Key Words: Attraction-repulsion; well-posedness; asymptotic behavior; global solution; boundedness