×
Tian, Ya; Xie, Guoqing

Global boundedness and large time behavior in a signal-dependent motility system with nonlinear signal consumption. (English) Zbl 1531.35076

Z. Angew. Math. Phys. 75, No. 1, Paper No. 7, 16 p. (2024).
Summary: In this paper, we deal with the following system with nonlinear signal consumption \[ \begin{cases} {u_t} = \Delta (\gamma (v) u) + ru - \mu {u^\alpha}, & x \in \Omega, \; t>0,\\ {v_t} = \Delta v - {u^\beta } v, & x \in \Omega, \; t>0, \end{cases} \] under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \in \mathbb{R}^n\) \((n \geq 2)\). It shown that whenever \(r > 0\), \(\mu > 0\), \(\alpha > 2\), \(\beta > 0\) and \(\frac{\alpha}{\beta} > \frac{n+ 2}{2}\), then the original system will produce a global classical solution and the solution converges to equilibrium \[ \Bigg(\bigg(\frac{r}{\mu}\bigg)^{\frac{1}{\alpha - 1}}, 0\Bigg) \quad \text{ as } t \to \infty. \]

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35K51 Initial-boundary value problems for second-order parabolic systems
35K58 Semilinear parabolic equations
35K65 Degenerate parabolic equations
92C17 Cell movement (chemotaxis, etc.)

Keywords:

global existence; boundedness; signal-dependent motilities; large time behavior
Cite Review PDF
Full Text: DOI

References:

[1] Alikakos, N., \({L^p}\) bounds of solutions of reaction-diffusion equations, Commun. Part. Diff. Equ., 4, 827-868 (1979) · Zbl 0421.35009 · doi:10.1080/03605307908820113
[2] Amann, H., Dynamic theory of quasilinear parabolic equations, II: reaction-diffusion systems, Differ. Integral Equ., 3, 1, 13-75 (1990) · Zbl 0729.35062
[3] Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Function Spaces, Differential Operators and Nonlinear Analysis, Friedrichroda, 1992, Teubner-Texte Mathematics, vol. 133, Teubner, Stuttgart (1993) · Zbl 0810.35037
[4] Bai, X.; Winkler, M., Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana U. Math. J., 65, 553-583 (2016) · Zbl 1345.35117 · doi:10.1512/iumj.2016.65.5776
[5] Cao, X., Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Cont. Dyn. A., 35, 1891-1904 (2015) · Zbl 1515.35047 · doi:10.3934/dcds.2015.35.1891
[6] Fujie, K.; Jiang, J., Boundedness of classical solutions to a degenerate Keller-Segel type model with signal-dependent motilities, Acta Appl. Math., 176, 1-36 (2021) · Zbl 1478.35035 · doi:10.1007/s10440-021-00450-1
[7] Horstmann, D.; Wang, G., Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12, 159-177 (2001) · Zbl 1017.92006 · doi:10.1017/S0956792501004363
[8] Jin, H.; Wang, Z., Critical mass on the Keller-Segel system with signal-dependent motility, Proc. Am. Math. Soc., 148, 4855-4873 (2020) · Zbl 1448.35516 · doi:10.1090/proc/15124
[9] Jin, H.; Wang, Z.; Kim, Y., Boundedness, stabilization, and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 78, 1632-1657 (2018) · Zbl 1393.35100 · doi:10.1137/17M1144647
[10] Keller, E.; Segel, L., Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26, 399-415 (1970) · Zbl 1170.92306 · doi:10.1016/0022-5193(70)90092-5
[11] Lankeit, J.; Wang, Y., Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, Discrete Cont. Dyn. A, 37, 6099-6121 (2017) · Zbl 1375.35575 · doi:10.3934/dcds.2017262
[12] Lady’zhenskaya, OA; Solonnikov, VA; Ural’ceva, NN, Linear and Quasilinear Equations of Parabolic Type (1968), Providence: AMS, Providence · Zbl 0174.15403 · doi:10.1090/mmono/023
[13] Li, D.; Zhao, J., Global boundedness and large time behavior of solutions to a chemotaxis-consumption system with signal-dependent motility, Z. Angew. Math. Phys., 72, 1-20 (2021) · Zbl 1467.92040
[14] Li, G.; Wang, L., Boundedness in a taxis-consumption system involving signal-dependent motilities and concurrent enhancement of density-determined diffusion and cross-diffusion, Z. Angew. Math. Phys., 74, 92 (2023) · Zbl 1518.35454 · doi:10.1007/s00033-023-01983-1
[15] Li, G.; Winkler, M., Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities, Appl. Anal., 8, 1-20 (2023)
[16] Li, G.; Winkler, M., Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities, Commun. Math. Sci., 21, 2, 299-322 (2023) · Zbl 1518.35109 · doi:10.4310/CMS.2023.v21.n2.a1
[17] Li, X.; Wang, L.; Pan, X., Boundedness and stabilization in the chemotaxis consumption model with signal-dependent motility, Z. Angew. Math. Phys., 72, 1-18 (2021) · Zbl 1471.92063
[18] Liu, Z.; Xu, J., Large time behavior of solutions for density-suppressed motility system in higher dimensions, J. Math. Anal. Appl., 475, 1596-1613 (2019) · Zbl 1416.35277 · doi:10.1016/j.jmaa.2019.03.033
[19] Lv, W., Global existence for a class of chemotaxis-consumption systems with signal-dependent motility and generalized logistic source, Nonlinear Anal. Real World Appl., 56, 103160 (2020) · Zbl 1471.35179 · doi:10.1016/j.nonrwa.2020.103160
[20] Lv, W.; Wang, Q., An n-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization, Proc. Roy. Soc. Edinb. A, 151, 821-841 (2021) · Zbl 1467.35324 · doi:10.1017/prm.2020.38
[21] Lv, W.; Wang, Q., Global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term, Evol. Equ. Control Theory, 10, 25 (2021) · Zbl 1480.35005 · doi:10.3934/eect.2020040
[22] Nagai, T.; Senba, T.; Yoshida, K., Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, 40, 411-433 (1997) · Zbl 0901.35104
[23] Osaki, K.; Yagi, A., Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkc. Ekvacioj, 44, 441-469 (2001) · Zbl 1145.37337
[24] Tao, Y., Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381, 521-529 (2011) · Zbl 1225.35118 · doi:10.1016/j.jmaa.2011.02.041
[25] Tao, X.; Fang, Z., Global boundedness and stability in a density-suppressed motility model with generalized logistic source and nonlinear signal production, Z. Angew. Math. Phys., 73, 1-19 (2022) · Zbl 1491.35028 · doi:10.1007/s00033-022-01775-z
[26] Tao, Y.; Winkler, M., Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., 252, 2520-2543 (2012) · Zbl 1268.35016 · doi:10.1016/j.jde.2011.07.010
[27] Tao, Y.; Winkler, M., Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Mod. Meth. Appl. Sci., 27, 1645-1683 (2017) · Zbl 1516.35092 · doi:10.1142/S0218202517500282
[28] Tao, Y.; Winkler, M., Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension, J. Differ. Equ., 343, 390-418 (2023) · Zbl 1505.35340 · doi:10.1016/j.jde.2022.10.022
[29] Tello, J.; Winkler, M., A chemotaxis system with logistic source, Commun. Part. Diff. Equ., 32, 849-877 (2007) · Zbl 1121.37068 · doi:10.1080/03605300701319003
[30] Wang, L., Improvement of conditions for boundedness in a chemotaxis consumption system with density-dependent motility, Appl. Math. Lett., 125, 107724 (2022) · Zbl 1479.35010 · doi:10.1016/j.aml.2021.107724
[31] Winkler, M., Aggregation versus global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248, 2889-2905 (2010) · Zbl 1190.92004 · doi:10.1016/j.jde.2010.02.008
[32] Winkler, M., Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Part. Diff. Equ., 35, 1516-1537 (2010) · Zbl 1290.35139 · doi:10.1080/03605300903473426
[33] Winkler, M., Global asympototic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differ. Equ., 257, 1056-1077 (2014) · Zbl 1293.35048 · doi:10.1016/j.jde.2014.04.023
[34] Winkler, M., Application of the Moser-Trudinger inequality in the construction of global solutions to a strongly degenerate migration model, Bull. Math. Sci. (2023) · Zbl 1539.35140 · doi:10.1142/S1664360722500126
[35] Winkler, M., Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration-consumption interaction, Z. Angew. Math. Phys., 74, 32 (2023) · Zbl 1504.35181 · doi:10.1007/s00033-022-01925-3
[36] Winkler, M., Can simultaneous density-determined enhancement of diffusion and cross-diffusion foster boundedness in Keller-Segel type systems involving signal-dependent motilities?, Nonlinearity, 33, 6590-6623 (2020) · Zbl 1454.35224 · doi:10.1088/1361-6544/ab9bae
[37] Yoon, C.; Kim, Y., Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Appl. Math., 149, 101-123 (2017) · Zbl 1398.35110 · doi:10.1007/s10440-016-0089-7
[38] Zhang, Q.; Li, Y., Stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant, J. Math. Phys., 56, 081506 (2015) · Zbl 1322.35060 · doi:10.1063/1.4929658
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.