In this work, we studied dynamic properties of classical solutions to the chemotaxis system in a two-dimensional smoothly bounded domain. By applying the Agmon-Douglis-Nirenberg $ L^{p} $ estimates on linear elliptic equations and the two-dimensional Gagliardo-Nirenberg interpolation inequality, we showed that $ \dfrac{\chi\tilde{\chi}}{\xi\tilde{\xi}}\geq1 $, which had nothing to do with $ \beta $ and $ \delta $, was sufficient to ensure the boundedness of the classical solutions. Meanwhile, by constructing Lyapunov function and the super-solution of the elliptic equations and employing the elliptic maximum principle, it was proved that the assumption $ \dfrac{\chi\tilde{\chi}}{\xi\tilde{\xi}} > \max\{\dfrac{\beta}{\delta}, \dfrac{\delta}{\beta}\} $ can guarantee the stabilization of the classical solutions. This study displayed the differences of both research methods and conclusions between the parabolic-elliptical-parabolic-elliptic system and fully parabolic system, and thus we extended the existing research ones.
Citation: Aichao Liu, Junhong Xiong, Lufang Mi. Boundedness and stabilization in a two-species parabolic-elliptic-parabolic-elliptic chemotaxis system with two chemicals[J]. AIMS Mathematics, 2026, 11(11): 35887-35901. doi: 10.3934/math.20261391
In this work, we studied dynamic properties of classical solutions to the chemotaxis system in a two-dimensional smoothly bounded domain. By applying the Agmon-Douglis-Nirenberg $ L^{p} $ estimates on linear elliptic equations and the two-dimensional Gagliardo-Nirenberg interpolation inequality, we showed that $ \dfrac{\chi\tilde{\chi}}{\xi\tilde{\xi}}\geq1 $, which had nothing to do with $ \beta $ and $ \delta $, was sufficient to ensure the boundedness of the classical solutions. Meanwhile, by constructing Lyapunov function and the super-solution of the elliptic equations and employing the elliptic maximum principle, it was proved that the assumption $ \dfrac{\chi\tilde{\chi}}{\xi\tilde{\xi}} > \max\{\dfrac{\beta}{\delta}, \dfrac{\delta}{\beta}\} $ can guarantee the stabilization of the classical solutions. This study displayed the differences of both research methods and conclusions between the parabolic-elliptical-parabolic-elliptic system and fully parabolic system, and thus we extended the existing research ones.
| [1] |
A. Liu, B. Dai, Y. Chen, Boundedness in a two-species attraction-repulsion chemotaxis system with two chemicals, Discrete Cont. Dyn. Syst. B, 27 (2022), 6037–6062. http://dx.doi.org/10.3934/dcdsb.2021306 doi: 10.3934/dcdsb.2021306
|
| [2] |
A. Liu, B. Dai, Blow-up vs boundedness in a two-species attraction-repulsion chemotaxis system with two chemicals, J. Math. Phys., 62 (2021), 111508. http://dx.doi.org/10.1063/5.0069180 doi: 10.1063/5.0069180
|
| [3] |
A. Liu, L. Mi, Asymptotic stability of the two-dimensional two-species attraction-repulsion chemotaxis system with two chemicals, Discrete Cont. Dyn. Syst. B, 31 (2026), 17–31. http://dx.doi.org/10.3934/dcdsb.2025101 doi: 10.3934/dcdsb.2025101
|
| [4] |
Y. Tao, Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1–36. http://dx.doi.org/10.1142/S0218202512500443 doi: 10.1142/S0218202512500443
|
| [5] |
N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663–1763. http://dx.doi.org/10.1142/S021820251550044X doi: 10.1142/S021820251550044X
|
| [6] |
K. Lin, T. Xiang, On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop, Calc. Var., 59 (2020), 108. http://dx.doi.org/10.1007/s00526-020-01777-7 doi: 10.1007/s00526-020-01777-7
|
| [7] |
S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅰ, Commun. Pure Appl. Math., 12 (1959), 623–727. http://dx.doi.org/10.1002/cpa.3160120405 doi: 10.1002/cpa.3160120405
|
| [8] |
S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅱ, Commun. Pure Appl. Math., 17 (1964), 35–92. http://dx.doi.org/10.1002/cpa.3160170104 doi: 10.1002/cpa.3160170104
|
| [9] |
H. Y. Jin, Z. A. Wang, Global stabilization of the full attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. A, 40 (2020), 3509–3527. http://dx.doi.org/10.3934/dcds.2020027 doi: 10.3934/dcds.2020027
|
| [10] |
Y. Tao, M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229–4250. http://dx.doi.org/10.1137/15M1014115 doi: 10.1137/15M1014115
|
| [11] |
D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, 2 Eds., Berlin; New York: Springer-Verlag, 1983. |
| [12] |
K. Lin, C. Mu, Global existence and convergence to steady states for an attraction-repulsion chemotaxis system, Nonlinear Anal. Real Word Appl., 31 (2016), 630–642. http://dx.doi.org/10.1016/j.nonrwa.2016.03.012 doi: 10.1016/j.nonrwa.2016.03.012
|
Aichao Liu, Junhong Xiong, Lufang Mi. Boundedness and stabilization in a two-species parabolic-elliptic-parabolic-elliptic chemotaxis system with two chemicals[J]. AIMS Mathematics, 2026, 11(11): 35887-35901. doi: 10.3934/math.20261391
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