Department of Basic Science
Jilin Jianzhu University
Changchun 130118, China
email: liuling2004@sohu.com
Ling Liu
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
DOI: 10.58997/ejde.2025.26
Show me the PDF file (405 KB), TEX file for this article.
|
Ling Liu Department of Basic Science Jilin Jianzhu University Changchun 130118, China email: liuling2004@sohu.com |
|---|
Return to the EJDE web page