The effects of cross-diffusion and logistic source on the boundedness of solutions to a pursuit-evasion model

Chang-Jian Wang; Zi-Han Zheng; Chang-Jian Wang; Zi-Han Zheng

doi:10.3934/era.2023170

Electronic Research Archive

2023, Volume 31, Issue 6: 3362-3380. doi: 10.3934/era.2023170

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Research article

The effects of cross-diffusion and logistic source on the boundedness of solutions to a pursuit-evasion model

Chang-Jian Wang ^,,
Zi-Han Zheng

School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

Received: 26 January 2023 Revised: 21 March 2023 Accepted: 04 April 2023 Published: 17 April 2023

We study the following quasilinear pursuit-evasion model:

$ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \Delta u-\chi\nabla \cdot (u(u+1)^{\alpha}\nabla w)+u(\lambda_{1}-\mu_{1}u^{r_{1}-1}+ av),\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] v_{t} = \Delta v+\xi\nabla \cdot(v(v+1)^{\beta}\nabla z)+v(\lambda_{2}-\mu_{2}v^{r_{2}-1}-bu), \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0 = \Delta w-w+v, \ &\ \ x\in \Omega, \ t>0 ,\\[2.5mm] 0 = \Delta z-z+u,\ &\ \ x\in \Omega, \ t>0 , \end{array} \right. \end{equation*} $

in a smooth and bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1), $ where $ a, b, \chi, \xi, \lambda_{1}, \lambda_{2}, \mu_{1}, \mu_{2} > 0, $ $ \alpha, \beta \in\mathbb{R}, $ and $ r_{1}, r_{2} > 1. $ When $ r_{1} > \max\{1, 1+\alpha\}, r_{2} > \max\{1, 1+\beta\}, $ it has been proved that if $ \min\{(r_{1}-1)(r_{2}-\beta-1), (r_{1}-\alpha-1)(r_{2}-\beta-1)\} > \frac{(n-2)_{+}}{n}, $ then for some suitable nonnegative initial data $ u_{0} $ and $ v_{0}, $ the system admits a unique globally classical solution which is bounded in $ \Omega\times(0, \infty) $.
- boundedness criteria,
- pursuit-evasion model,
- cross-diffusion,
- logistic source
Citation: Chang-Jian Wang, Zi-Han Zheng. The effects of cross-diffusion and logistic source on the boundedness of solutions to a pursuit-evasion model[J]. Electronic Research Archive, 2023, 31(6): 3362-3380. doi: 10.3934/era.2023170

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Abstract

We study the following quasilinear pursuit-evasion model:

$ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \Delta u-\chi\nabla \cdot (u(u+1)^{\alpha}\nabla w)+u(\lambda_{1}-\mu_{1}u^{r_{1}-1}+ av),\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] v_{t} = \Delta v+\xi\nabla \cdot(v(v+1)^{\beta}\nabla z)+v(\lambda_{2}-\mu_{2}v^{r_{2}-1}-bu), \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0 = \Delta w-w+v, \ &\ \ x\in \Omega, \ t>0 ,\\[2.5mm] 0 = \Delta z-z+u,\ &\ \ x\in \Omega, \ t>0 , \end{array} \right. \end{equation*} $

in a smooth and bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1), $ where $ a, b, \chi, \xi, \lambda_{1}, \lambda_{2}, \mu_{1}, \mu_{2} > 0, $ $ \alpha, \beta \in\mathbb{R}, $ and $ r_{1}, r_{2} > 1. $ When $ r_{1} > \max\{1, 1+\alpha\}, r_{2} > \max\{1, 1+\beta\}, $ it has been proved that if $ \min\{(r_{1}-1)(r_{2}-\beta-1), (r_{1}-\alpha-1)(r_{2}-\beta-1)\} > \frac{(n-2)_{+}}{n}, $ then for some suitable nonnegative initial data $ u_{0} $ and $ v_{0}, $ the system admits a unique globally classical solution which is bounded in $ \Omega\times(0, \infty) $.

References

[1]	E. Keller, L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399–415. https://doi.org/10.1016/0022-5193(70)90092-5 doi: 10.1016/0022-5193(70)90092-5
[2]	T. Ciéslak, M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057–1076. https://doi.org/10.1088/0951-7715/21/5/009 doi: 10.1088/0951-7715/21/5/009
[3]	D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. DMV, 105 (2003), 103–165.
[4]	R. Kowalczyk, Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566–588. https://doi.org/10.1016/j.jmaa.2004.12.009 doi: 10.1016/j.jmaa.2004.12.009
[5]	K. Osaki, A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441–469.
[6]	H. Gajewski, K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77–114. https://doi.org/10.1002/mana.19981950106 doi: 10.1002/mana.19981950106
[7]	D. Horstmann, G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159–177. https://doi.org/10.1017/S0956792501004363 doi: 10.1017/S0956792501004363
[8]	T. Nagai, T. Senba, K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411–433.
[9]	T. Senba, T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349–367. https://doi.org/10.4310/MAA.2001.v8.n2.a9 doi: 10.4310/MAA.2001.v8.n2.a9
[10]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equations, 248 (2010), 2889–2905. https://doi.org/10.1016/j.jde.2010.02.008 doi: 10.1016/j.jde.2010.02.008
[11]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748–767. https://doi.org/10.1016/j.matpur.2013.01.020 doi: 10.1016/j.matpur.2013.01.020
[12]	D. Liu, Y. Tao, Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chin. Univ. Ser. B, 31 (2016), 379–388. https://doi.org/10.1007/s11766-016-3386-z doi: 10.1007/s11766-016-3386-z
[13]	M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031–2056. https://doi.org/10.1088/1361-6544/aaaa0e doi: 10.1088/1361-6544/aaaa0e
[14]	J. I. Tello, M. Winkler, Chemotaxis system with logistic source, Comm. Partial Differ. Equations, 32 (2007), 849–877. https://doi.org/10.1080/03605300701319003 doi: 10.1080/03605300701319003
[15]	Z. Wang, T. Xiang, A class of chemotaxis systems with growth source and nonlinear secretion, preprient, arXiv: 1510.07204.
[16]	M. Winkler, Chemotaxis with logistic source: very weak global solutions and boundedness properties, J. Math. Anal. Appl., 348 (2008), 708–729. https://doi.org/10.1016/j.jmaa.2008.07.071 doi: 10.1016/j.jmaa.2008.07.071
[17]	G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197–212. https://doi.org/10.1016/j.jmaa.2016.02.069 doi: 10.1016/j.jmaa.2016.02.069
[18]	M. Winkler, The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in $L^{1}$, Adv. Nonlinear Anal., 9 (2020), 526–566. https://doi.org/10.1515/anona-2020-0013 doi: 10.1515/anona-2020-0013
[19]	K. Painter, T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501–543.
[20]	M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse? Math, Methods Appl. Sci., 33 (2010), 12–24. https://doi.org/10.1002/mma.1146 doi: 10.1002/mma.1146
[21]	Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equations, 252 (2012), 692–715. https://doi.org/10.1016/j.jde.2011.08.019 doi: 10.1016/j.jde.2011.08.019
[22]	K. Lin, C. Mu, H. Zhong, A blow-up result for a quasilinear chemotaxis system with logistic source in higher dimensions, J. Math. Anal. Appl., 464 (2018), 435–455. https://doi.org/10.1016/j.jmaa.2018.04.015 doi: 10.1016/j.jmaa.2018.04.015
[23]	X. Cao, Y. Tao, Boundedness and stabilization enforced by mild saturation of taxis in a producer-scrounger model, Nonlinear Anal. Real World Appl., 57 (2021), 103189. https://doi.org/10.1016/j.nonrwa.2020.103189 doi: 10.1016/j.nonrwa.2020.103189
[24]	X. Li, Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503–3531. https://doi.org/10.3934/dcds.2015.35.3503 doi: 10.3934/dcds.2015.35.3503
[25]	L. Wang, C. Mu, P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differ. Equations, 256 (2014), 1847–1872. https://doi.org/10.1016/j.jde.2013.12.007 doi: 10.1016/j.jde.2013.12.007
[26]	L. Wang, Y. Li, C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789–802. https://doi.org/10.3934/dcds.2014.34.789 doi: 10.3934/dcds.2014.34.789
[27]	Q. Zhang, Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473–2484. https://doi.org/10.1007/s00033-015-0532-z doi: 10.1007/s00033-015-0532-z
[28]	C. Wang, L. Zhao, X. Zhu, A blow-up result for attraction-repulsion system with nonlinear signal production and generalized logistic source, J. Math. Anal. Appl., 518 (2023), 126679. https://doi.org/10.1016/j.jmaa.2022.126679 doi: 10.1016/j.jmaa.2022.126679
[29]	C. Wang, Y. Yang, Boundedness criteria for the quasilinear attraction-repulsion chemotaxis system with nonlinear signal production and logistic source, Electron. Res. Arch., 31 (2023), 299–318. https://doi.org/10.3934/era.2023015 doi: 10.3934/era.2023015
[30]	C. Wang, Y. Zhu, X. Zhu, Long time behavior of the solution to a chemotaxis system with nonlinear indirect signal production and logistic source, Electron. J. Qual. Theory Differ. Equations, 11 (2023), 1–21. https://doi.org/10.14232/ejqtde.2023.1.11 doi: 10.14232/ejqtde.2023.1.11
[31]	Y. Tao, M. Winkler, A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation, J. Funct. Anal., 281 (2021), 109069. https://doi.org/10.1016/j.jfa.2021.109069 doi: 10.1016/j.jfa.2021.109069
[32]	Y. Tao, M. Winkler, Existence theory and qualitative analysis for a fully cross-diffusive predator-prey system, SIAM J. Math. Anal., 54 (2022), 4806–4864. https://doi.org/10.1137/21M1449841 doi: 10.1137/21M1449841
[33]	A. Chakraborty, M. Singh, D. Lucy, P. Ridland, Predator-prey model with prey-taxis and diffusion, Math. Comput. Modell., 46 (2007), 482–498. https://doi.org/10.1016/j.mcm.2006.10.010 doi: 10.1016/j.mcm.2006.10.010
[34]	J. I. Tello, D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129–2162. https://doi.org/10.1142/s0218202516400108 doi: 10.1142/s0218202516400108
[35]	I. Ahn, C. Yoon, Global solvability of prey-predator models with indirect predator-taxis, Z. Angew. Math. Phys., 72 (2021), 1–20. https://doi.org/10.1007/s00033-020-01461-y doi: 10.1007/s00033-020-01461-y
[36]	H. Jin, Z. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, Eur. J. Appl. Math., 32 (2021), 652–682. https://doi.org/10.1017/s0956792520000248 doi: 10.1017/s0956792520000248
[37]	G. Ren, Y. Shi, Global boundedness and stability of solutions for prey-taxis model with handling and searching predators, Nonlinear Anal. Real World Appl., 60 (2021), 103306. https://doi.org/10.1016/j.nonrwa.2021.103306 doi: 10.1016/j.nonrwa.2021.103306
[38]	T. Goudon, L. Urrutia, Analysis of kinetic and macroscopic models of pursuit-evasion dynamics, Commun. Math. Sci., 14 (2016), 2253–2286. https://doi.org/10.4310/CMS.2016.v14.n8.a7 doi: 10.4310/CMS.2016.v14.n8.a7
[39]	P. Amorim, B. Telch, L. M. Villada, A reaction-diffusion predator-prey model with pursuit, evasion, and nonlocal sensing, Math. Biosci. Eng., 16 (2019), 5114–5145. https://doi.org/10.3934/mbe.2019257 doi: 10.3934/mbe.2019257
[40]	G. Li, Y. Tao, M. Winkler, Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 4383. https://doi.org/10.3934/dcdsb.2020102 doi: 10.3934/dcdsb.2020102
[41]	J. Zheng, P. Zhang, Blow-up prevention by logistic source an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction, J. Math. Anal. Appl., 519(2023), 126741. https://doi.org/10.1016/j.jmaa.2022.126741 doi: 10.1016/j.jmaa.2022.126741
[42]	C. Liu, B. Liu, Boundedness and asymptotic behavior in a predator-prey model with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), 4855–4874. https://doi.org/10.3934/dcdsb.2021255 doi: 10.3934/dcdsb.2021255
[43]	D. Qi, Y. Ke, Large time behavior in a predator-prey system with indirect pursuit-evasion interaction, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), 4531–4549. https://doi.org/10.3934/dcdsb.2021240 doi: 10.3934/dcdsb.2021240
[44]	P. Amorim, B. Telch, A chemotaxis predator-prey model with indirect pursuit-evasion dynamics and parabolic signal, J. Math. Anal. Appl., 500 (2021), 125128. https://doi.org/10.1016/j.jmaa.2021.125128 doi: 10.1016/j.jmaa.2021.125128
[45]	B. Telch, Global boundedness in a chemotaxis quasilinear parabolic predator-prey system with pursuit-evasion, Nonlinear Anal. Real World Appl., 59 (2021), 103269. https://doi.org/10.1016/j.nonrwa.2020.103269 doi: 10.1016/j.nonrwa.2020.103269
[46]	B. Telch, A parabolic-quasilinear predator-prey model under pursuit-evasion dynamics, J. Math. Anal. Appl., 514 (2022), 126276. https://doi.org/10.1016/j.jmaa.2022.126276 doi: 10.1016/j.jmaa.2022.126276
[47]	Y. Wang, A quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type with logistic source, J. Math. Anal. Appl., 441 (2016), 259–292. https://doi.org/10.1016/j.jmaa.2016.03.061 doi: 10.1016/j.jmaa.2016.03.061
[48]	M. Winkler, K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044–1064. https://doi.org/10.1016/j.na.2009.07.045 doi: 10.1016/j.na.2009.07.045
[49]	L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., Ⅲ. Ser., 13 (1959), 115–162.
[50]	H. Brézis, W. A. Strauss, Semi-linear second-order elliptic equations in $L^{1}, $ J. Math. Soc. Jpn., 25 (1973), 565–590. https://doi.org/10.2969/jmsj/02540565 doi: 10.2969/jmsj/02540565

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