Dynamics of Partial Differential Equations

Volume 21 (2024)

Number 1

On a parabolic-elliptic Keller–Segel system with nonlinear signal production and nonlocal growth term

Pages: 61 – 76

DOI: https://dx.doi.org/10.4310/DPDE.2024.v21.n1.a3

Author

Pan Zheng (School of Science, Chongqing University of Posts & Telecom., Chongqing, China; Department of Mathematics, Chinese University of Hong Kong; and School of Maths. & Statistics, Yunnan University, Kunming, China)

Abstract

and nonlocal growth term\[\begin{cases}u_t = \Delta u - \chi \nabla \cdot (u^m \nabla v) + u \Biggl( a_0 - a_1 u^\alpha + a_2 \displaystyle \int_\Omega u^\sigma dx \Biggr) & (x, t) \in \Omega \times (0,\infty) \; , \\0=\Delta - v + u^\gamma , & (x, t) \in \Omega \times (0,\infty) \; , \\\end{cases}\]under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subset \mathbb{R}^n (n \geq 1)$, where $\chi \in \mathbb{R}, m, \gamma \geq 1$ and $a_0, a_1, a_2, \alpha \gt 0$.

• When $\chi \gt 0$, the solution of the above system is global and uniformly bounded, if the parameters satisfy certain suitable assumptions.

• When $\chi \gt 0$, the system possesses a globally bounded classical solution, provided that $a_1 \gt a_2 \lvert \Omega \rvert$.

These results indicate that the repulsive mechanism plays a crucial role in ensuring the global boundedness of solutions. In addition, the paper derives the large time behavior of globally bounded solutions for the chemo-attractive or chemo-repulsive system by constructing energy functionals.

Keywords

nonlocal growth term, chemo-attraction, chemo-repulsion, boundedness, large time behavior

2010 Mathematics Subject Classification

Primary 35K15, 35K55. Secondary 92C17.

Received 2 April 2022

Published 7 November 2023