GLOBAL BOUNDEDNESS IN A QUASILINEAR CHEMOTAXIS-CONSUMPTION SYSTEM WITH SIGNAL-DEPENDENT MOTILITY AND SUPER-QUADRATIC DAMPING

Chi Xu

doi:10.11948/20240271

2025 Volume 15 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(5): 2570-2588

Next Article Previous Article

Chi Xu. GLOBAL BOUNDEDNESS IN A QUASILINEAR CHEMOTAXIS-CONSUMPTION SYSTEM WITH SIGNAL-DEPENDENT MOTILITY AND SUPER-QUADRATIC DAMPING[J]. Journal of Applied Analysis & Computation, 2025, 15(5): 2570-2588. doi: 10.11948/20240271

Citation:

Chi Xu. GLOBAL BOUNDEDNESS IN A QUASILINEAR CHEMOTAXIS-CONSUMPTION SYSTEM WITH SIGNAL-DEPENDENT MOTILITY AND SUPER-QUADRATIC DAMPING[J]. Journal of Applied Analysis & Computation, 2025, 15(5): 2570-2588. doi: 10.11948/20240271

GLOBAL BOUNDEDNESS IN A QUASILINEAR CHEMOTAXIS-CONSUMPTION SYSTEM WITH SIGNAL-DEPENDENT MOTILITY AND SUPER-QUADRATIC DAMPING

Chi Xu^1, ,

1.
School of Mathematics and Big Data, Anhui University of Science and Technology, 232001 Huainan, China

Corresponding author: Email: XuChi1993@126.com(C. Xu)
Fund Project: The author was supported by the Key Program of the University Natural Science Research Fund of Anhui Provence (2022AH050826)

Abstract

Abstract

In this paper, we consider a quasilinear chemotaxis-consumption model
$ \left\{ \begin{array}{lll} u_t=\Delta (v^{\alpha}u^{m})+ru-\mu u^{l},\quad &x\in\Omega,\; t>0,\\ v_t=\Delta v-uv, &x\in\Omega,\; t>0 \end{array} \right. $
within a smoothly bounded domain $ \Omega\subset\mathbb{R}^n $ under homogeneous Neumann boundary conditions, where the parameters $ \alpha, r, \mu>0 $ and $ l, m>1 $. For any sufficiently regular initial data and parameters $ l, m>1 $ with $ l>m+1 $, it is shown that the aforementioned system possesses at least one global weak solution with a boundedness property
$ \|u(\cdot,t)\|_{L^{p}(\Omega)}+\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}\leq C $
for all $ p\geq 2 $ and $ t>0 $. This finding indicates the regularizing effect of super-quadratic damping of a logistic-type source under strong degeneracy of signal-dependent motility, even though the cross-diffusion is simultaneously enhanced.
- Chemotaxis /
- weak solution /
- degenerate diffusion /
- boundedness
MSC: 35A01, 35K55, 35K65, 92C17, 35Q92

FullText(HTML)

References

[1]	N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 2015, 25(9), 1663–1763. doi: 10.1142/S021820251550044X CrossRef Google Scholar
[2]	Z. Chen and G. Li, Global weak solution in a singular taxis-type system with signal consumption, Nonlinear Anal., 2024, 78, 104073. doi: 10.1016/j.nonrwa.2024.104073 CrossRef Google Scholar
[3]	L. Desvillettes, P. Laurençot, A. Trescases and M. Winkler, Weak solutions to triangular cross-diffusion systems modeling chemotaxis with local sensing, Nonlinear Anal., 2023, 226, 113153. doi: 10.1016/j.na.2022.113153 CrossRef Google Scholar
[4]	X. Fu, L. Tang, C. Liu, J. Huang, T. Hwa and P. Lenz, Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 2012, 39, 1185–1284. Google Scholar
[5]	K. Fujie and J. Jiang, Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities, Calc. Var. Partial Differential Equations, 2021, 60, 1–37. doi: 10.1007/s00526-020-01865-8 CrossRef Google Scholar
[6]	K. Fujie and T. Senba, Global boundedness of solutions to a parabolic-parabolic chemotaxis system with local sensing in higher dimensions, Nonlinearity, 2022, 35, 3777–3811. doi: 10.1088/1361-6544/ac6659 CrossRef Google Scholar
[7]	K. Fujie and T. Senba, Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions, Nonlinear Anal., 2022, 222, 112987. doi: 10.1016/j.na.2022.112987 CrossRef Google Scholar
[8]	H. Jin, Y. Kim and Z. Wang, Boundedness, stabilization and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 2018, 78(3), 1632–1657. doi: 10.1137/17M1144647 CrossRef Google Scholar
[9]	H. Jin and Z. Wang, Critical mass on the Keller-Segel system with signal-dependent motility, Proc. Amer. Math. Soc., 2020, 148(11), 4855–4873. doi: 10.1090/proc/15124 CrossRef Google Scholar
[10]	J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion, J. Differential Equations, 2017, 262, 4052–4084. doi: 10.1016/j.jde.2016.12.007 CrossRef Google Scholar
[11]	J. Lankeit and M. Winkler, Depleting the signal: Analysis of chemotaxis-consumption models–A survey, Stud. Appl. Math., 2023, 151(4), 1197–1229. doi: 10.1111/sapm.12625 CrossRef Google Scholar
[12]	P. Laurençot, Large time convergence for a chemotaxis model with degenerate local sensing and consumption, Bull. Korean Math. Soc., 2024, 61(2), 479–488. Google Scholar
[13]	D. Li and J. Zhan, Global boundedness and large time behavior of solutions to a chemotaxis-consumption system with signal-dependent motility, Z. Angew. Math. Phys., 2021, 72, 57. doi: 10.1007/s00033-021-01493-y CrossRef Google Scholar
[14]	G. Li and Y. Lou, Roles of density-related diffusion and signal-dependent motilities in a chemotaxis-consumption system, Calc. Var. and Partial Differential Equations, 2024, 63, 195. doi: 10.1007/s00526-024-02802-9 CrossRef Google Scholar
[15]	G. Li and L. Wang, Boundedness in a taxis-consumption system involving signal-dependent motilities and concurrent enhancement of density-determined diffusion and cross-diffusion, Z. Angew. Math. Phys., 2023, 74, 92. doi: 10.1007/s00033-023-01983-1 CrossRef Google Scholar
[16]	G. Li and M. Winkler, Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities, Appl. Anal., 2023, 1–20. Google Scholar
[17]	G. Li and M. Winkler, Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities, Commum. Math. Sci., 2023, 21, 299–322. doi: 10.4310/CMS.2023.v21.n2.a1 CrossRef Google Scholar
[18]	G. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl., 1987, 148, 319–351. Google Scholar
[19]	W. Lv, Global existence for a class of chemotaxis-consumption systems with signal-dependent motility and generalized logistic source, Nonlinear Anal., 2020, 56, 103160. doi: 10.1016/j.nonrwa.2020.103160 CrossRef Google Scholar
[20]	W. Lv and Q. Wang, An $n$-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: Global existence and asymptotic stabilization, Proc. Roy. Soc. Edinburgh Sect. A, 2021, 151(2), 821–841. doi: 10.1017/prm.2020.38 CrossRef Google Scholar
[21]	W. Lv and Z. Wang, Logistic damping effect in chemotaxis models with density-suppressed motility, Adv. Nonlinear Anal., 2023, 12(1), 336–355. Google Scholar
[22]	M. Ma, R. Peng and Z. Wang, Stationary and non-stationary patterns of the density-suppressed motility model, Phys. D, 2020, 402, 132259. doi: 10.1016/j.physd.2019.132259 CrossRef Google Scholar
[23]	M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 1993, 103(1), 146–178. doi: 10.1006/jdeq.1993.1045 CrossRef Google Scholar
[24]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subscritical sensitivity, J. Differential Equations, 2012, 252, 692–715. doi: 10.1016/j.jde.2011.08.019 CrossRef Google Scholar
[25]	Y. Tao and M. Winkler, Global solution to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domain of arbitrary dimension, J. Differential Equations, 2023, 343, 390–418. doi: 10.1016/j.jde.2022.10.022 CrossRef Google Scholar
[26]	J. Wang and M. Wang, Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth, J. Math. Phys., 2019, 60, 011507. doi: 10.1063/1.5061738 CrossRef Google Scholar
[27]	L. Wang, Improvement of conditions for boundedness in a chemotaxis consumption system with density-dependent motility, Appl. Math. Lett., 2022, 125, 107724. doi: 10.1016/j.aml.2021.107724 CrossRef Google Scholar
[28]	L. Wang, Global dynamics for a chemotaxis consumption system with signal-dependent motility and logistic source, J. Differential Equations, 2023, 348, 191–222. doi: 10.1016/j.jde.2022.12.004 CrossRef Google Scholar
[29]	L. Wang and R. Huang, Global classical solution to a chemotaxis consumption model involving singularly signal-dependent motility and logistic source, Nonlinear Anal., Real World Appl., 2024, 80, 104174. doi: 10.1016/j.nonrwa.2024.104174 CrossRef Google Scholar
[30]	M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 2016, 26, 987–1024. doi: 10.1142/S0218202516500238 CrossRef Google Scholar
[31]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 2010, 248, 2889–2905. doi: 10.1016/j.jde.2010.02.008 CrossRef Google Scholar
[32]	M. Winkler, Small-signal solutions of a two-dimensional doubly degenerate taxis system modeling bacterial motion in nutrient-poor environment, Nonlinear Anal. Real World Appl., 2022, 63, 103407. doi: 10.1016/j.nonrwa.2021.103407 CrossRef Google Scholar
[33]	M. Winkler, Approaching logarithmic singularities in quasilinear chemotaxis-consum-ption systems with signal-dependent sensitivities, Discrete Contin. Dyn. Ser B., 2022, 27(11), 6565–6587. doi: 10.3934/dcdsb.2022009 CrossRef Google Scholar
[34]	M. Winkler, Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migration-consumption interation, Nonlinearity, 2023, 36(8), 4438–4469. doi: 10.1088/1361-6544/ace22e CrossRef Google Scholar
[35]	M. Winkler, A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2024, 41, 95–127. Google Scholar
[36]	M. Winkler, Can simultaneous density-determined enhancement of diffusion and cross-diffusion foster boundedness in Keller-Segel type system involving signal-dependent motilities?, Nonlinearity, 2020, 33(12), 6590–6632. doi: 10.1088/1361-6544/ab9bae CrossRef Google Scholar