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


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)


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.


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

2010 Mathematics Subject Classification

Primary 35K15, 35K55. Secondary 92C17.

This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11601053, 12271064), by the Science and Technology Research Project of Chongqing Municipal Education Commission (Grant No. KJZD-K202200602), by the Natural Science Foundation of Chongqing (Grant No. CSTB2023NSCQ-MSX0099), by the Hong Kong Scholars Program (Grant Nos. XJ2021042, 2021-005), and by the Young Hundred Talents Program of CQUPT in 2022–2024.

Received 2 April 2022

Published 7 November 2023