Blow-up and boundedness in quasilinear attraction-repulsion systems with nonlinear signal production

Ruxi Cao; Zhongping Li; Ruxi Cao; Zhongping Li

doi:10.3934/mbe.2023243

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 3: 5243-5267. doi: 10.3934/mbe.2023243

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Blow-up and boundedness in quasilinear attraction-repulsion systems with nonlinear signal production

Ruxi Cao ,
Zhongping Li ^,

College of Mathematics and Information, China West Normal University, Nanchong 637009, China

Academic Editor: Zhi-An Wang

Received: 19 October 2022 Revised: 08 December 2022 Accepted: 13 December 2022 Published: 10 January 2023

In this paper, we consider the quasilinear parabolic-elliptic-elliptic attraction-repulsion system

$ \begin{equation} \nonumber \left\{ \begin{split} &u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w),&\qquad &x\in\Omega,\,t>0, \\ & 0 = \Delta v-\mu_{1}(t)+f_{1}(u),&\qquad &x\in\Omega,\,t>0, \\ &0 = \Delta w-\mu_{2}(t)+f_{2}(u),&\qquad &x\in\Omega,\,t>0 \end{split} \right. \end{equation} $

under homogeneous Neumann boundary conditions in a smooth bounded domain $ \Omega\subset\mathbb{R}^n, \ n\geq2 $. The nonlinear diffusivity $ D $ and nonlinear signal productions $ f_{1}, f_{2} $ are supposed to extend the prototypes

$ \begin{equation} \nonumber D(s) = (1+s)^{m-1},\ f_{1}(s) = (1+s)^{\gamma_{1}},\ f_{2}(s) = (1+s)^{\gamma_{2}},\ s\geq0,\gamma_{1},\gamma_{2}>0,m\in\mathbb{R}. \end{equation} $

We proved that if $ \gamma_{1} > \gamma_{2} $ and $ 1+\gamma_{1}-m > \frac{2}{n} $, then the solution with initial mass concentrating enough in a small ball centered at origin will blow up in finite time. However, the system admits a global bounded classical solution for suitable smooth initial datum when $ \gamma_{2} < 1+\gamma_{1} < \frac{2}{n}+m $.
- chemotaxis,
- quasilinear,
- attraction-repulsion,
- blow-up,
- boundedness
Citation: Ruxi Cao, Zhongping Li. Blow-up and boundedness in quasilinear attraction-repulsion systems with nonlinear signal production[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 5243-5267. doi: 10.3934/mbe.2023243

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Abstract

In this paper, we consider the quasilinear parabolic-elliptic-elliptic attraction-repulsion system

$ \begin{equation} \nonumber \left\{ \begin{split} &u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w),&\qquad &x\in\Omega,\,t>0, \\ & 0 = \Delta v-\mu_{1}(t)+f_{1}(u),&\qquad &x\in\Omega,\,t>0, \\ &0 = \Delta w-\mu_{2}(t)+f_{2}(u),&\qquad &x\in\Omega,\,t>0 \end{split} \right. \end{equation} $

under homogeneous Neumann boundary conditions in a smooth bounded domain $ \Omega\subset\mathbb{R}^n, \ n\geq2 $. The nonlinear diffusivity $ D $ and nonlinear signal productions $ f_{1}, f_{2} $ are supposed to extend the prototypes

$ \begin{equation} \nonumber D(s) = (1+s)^{m-1},\ f_{1}(s) = (1+s)^{\gamma_{1}},\ f_{2}(s) = (1+s)^{\gamma_{2}},\ s\geq0,\gamma_{1},\gamma_{2}>0,m\in\mathbb{R}. \end{equation} $

We proved that if $ \gamma_{1} > \gamma_{2} $ and $ 1+\gamma_{1}-m > \frac{2}{n} $, then the solution with initial mass concentrating enough in a small ball centered at origin will blow up in finite time. However, the system admits a global bounded classical solution for suitable smooth initial datum when $ \gamma_{2} < 1+\gamma_{1} < \frac{2}{n}+m $.

References

[1]	E. F. Keller, L. A. 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]	S. Ishida, K. Seki, T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differ. Equation, 256 (2014), 2993–3010. https://doi.org/10.1016/j.jde.2014.01.028 doi: 10.1016/j.jde.2014.01.028
[3]	Y. S. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equation, 252 (2012), 692–715. https://doi.org/10.1016/j.jde.2011.08.019 doi: 10.1016/j.jde.2011.08.019
[4]	Y. F. Wang, J. Liu, Boundedness in quasilinear fully parabolic Keller-Segel system with logistic source, Nonlinear Anal. Real World Appl., 38 (2017), 113–130. https://doi.org/10.1016/j.nonrwa.2017.04.010 doi: 10.1016/j.nonrwa.2017.04.010
[5]	J. S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differ. Equation, 259 (2015), 120–140. https://doi.org/10.1016/j.jde.2015.02.003 doi: 10.1016/j.jde.2015.02.003
[6]	A. Blanchet, P. Laurencot, The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\mathbb{R}^{d}, d\geq3$, Commun. Partial Differ. Equation, 38 (2013), 658–686. https://doi.org/10.1080/03605302.2012.757705 doi: 10.1080/03605302.2012.757705
[7]	T. Cie$\acute{s}$lak, C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabloic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differ. Equation, 252 (2012), 5832–5851. https://doi.org/10.1016/j.jde.2012.01.045 doi: 10.1016/j.jde.2012.01.045
[8]	T. Cie$\acute{s}$lak, C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differ. Equation, 258 (2015), 2080–2113. https://doi.org/10.1016/j.jde.2014.12.004 doi: 10.1016/j.jde.2014.12.004
[9]	T. Hashira, S. Ishida, T. Yokota, Finite-time blow-up for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differ. Equation, 264 (2018), 6459–6485. https://doi.org/10.1016/j.jde.2018.01.038 doi: 10.1016/j.jde.2018.01.038
[10]	P. Laurencot, N. Mizoguchi, Finite-time blowup for the parabolic-parabolic Keller-Segel system with nonlinear critical diffusion, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire., 34 (2017), 197–220. https://doi.org/10.1016/j.anihpc.2015.11.002 doi: 10.1016/j.anihpc.2015.11.002
[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]	V. Calvez, J. A. Carrillo, Volume effects in the Keller-Segel model:energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155–175. https://doi.org/10.1016/j.matpur.2006.04.002 doi: 10.1016/j.matpur.2006.04.002
[13]	T. Cie$\acute{s}$lak, P. Laurencot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system, C. R. Math. Acad. Sci. Paris., 347 (2009), 237–242. https://doi.org/10.1016/j.crma.2009.01.016 doi: 10.1016/j.crma.2009.01.016
[14]	T. Cie$\acute{s}$lak, 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
[15]	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
[16]	Y. Li, Finite-time blow-up in quasilinear parabolic-elliptic chemotaxis system with nonlinear signal production, Math. Anla. Appl., 480 (2019), 123376. https://doi.org/10.1016/j.jmaa.2019.123376 doi: 10.1016/j.jmaa.2019.123376
[17]	C. J. Wang, L. X. Zhao, X. C. 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
[18]	W. W. Wang, Y. X. Li, Boundedness and finite-time blow-up in a chemotaxis system with nonlinear signal production, Nonlinear Anal. RWA., 59 (2021), 103237. https://doi.org/10.1016/j.nonrwa.2020.103237 doi: 10.1016/j.nonrwa.2020.103237
[19]	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
[20]	P. Liu, J. P. Shi, Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst-Series B., 18 (2013), 2597–2625. https://doi.org/10.3934/dcdsb.2013.18.2597 doi: 10.3934/dcdsb.2013.18.2597
[21]	M. Luca, A. C. Ross, L. E. Keshet, Chemotactic signalling, microglia, and Alzheimer's disease senile plague: is there a connection?, Bull. Math. Biol., 65 (2003), 693–730. https://doi.org/10.1016/s0092-8240(03)00030-2 doi: 10.1016/s0092-8240(03)00030-2
[22]	B. Perthame, C. Schmeiser, M. Tang, N. Vauchelet, Travelling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion: existene and branching instabilites, Nonlinearity, 24 (2011), 1253–1270. https://doi.org/10.1088/0951-7715/24/4/012 doi: 10.1088/0951-7715/24/4/012
[23]	Y. S. Tao, Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Mod. Meth. Appl. Sci., 23 (2013), 1–36. https://doi.org/10.1142/s0218202512500443 doi: 10.1142/s0218202512500443
[24]	Y. Li, Y. X. Li, Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. RWA., 30 (2016), 170–183. https://doi.org/10.1016/j.nonrwa.2015.12.003 doi: 10.1016/j.nonrwa.2015.12.003
[25]	G. Viglialoro, Explicit lower bound of blow-up time for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 479 (2019), 1069–1077. https://doi.org/10.1016/j.jmaa.2019.06.067 doi: 10.1016/j.jmaa.2019.06.067
[26]	H. Y. Jin, Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differ. Equ., 260 (2016), 162–196. https://doi.org/10.1016/j.jde.2015.08.040 doi: 10.1016/j.jde.2015.08.040
[27]	H. Zhong, C. L. Mu, K. Lin, Global weak solution and boundedness in a three-dimensional competing chemotaxis, Discrete Contin. Dyn. Syst., 38 (2018), 3875–3898. https://doi.org/10.3934/dcds.2018168 doi: 10.3934/dcds.2018168
[28]	J. Liu, Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31–41. https://doi.org/10.1080/17513758.2011.571722 doi: 10.1080/17513758.2011.571722
[29]	H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463–1478. https://doi.org/10.1016/j.jmaa.2014.09.049 doi: 10.1016/j.jmaa.2014.09.049
[30]	H. Y. Jin, Z. A. Wang, Global stabilization of the full attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst., 40 (2020), 3509–3527. https://doi.org/10.3934/dcds.2020027 doi: 10.3934/dcds.2020027
[31]	H. Y. Jin, Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Meth. Appl. Sci., 38 (2015), 444–457. https://doi.org/10.1002/mma.3080 doi: 10.1002/mma.3080
[32]	Y. Chiyo, M. Marras, Y. Tanaka, T. Yokota, Blow-up phenomena in a parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with superlinear logistic degradation, Nonlinear Anal. RWA., 212 (2021), 112550. https://doi.org/10.1016/j.na.2021.112550 doi: 10.1016/j.na.2021.112550
[33]	Y. Chiyo, T. Yokota, Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system, Z. Angew. Math. Phys., 73 (2022), 1–21. https://doi.org/10.1007/s00033-022-01695-y doi: 10.1007/s00033-022-01695-y
[34]	X. C. Gao, J. Zhou, M. Tian, Global boundedness and asymptotic behavior for an attraction-repulsion chemotaxis system with logistic source, Acta Math. Sci. Ser. A (Chin. Ed.), 37 (2017), 113–121.
[35]	D. Li, C. L. Mu, K. Lin, L. C. Wang, Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, J. Math. Anla. Appl., 448 (2017), 914–936. https://doi.org/10.1016/j.jmaa.2016.11.036 doi: 10.1016/j.jmaa.2016.11.036
[36]	X. Li, Z. Y. Xiang, On an attraction-repulsion chemotaxis system with logistic source, IMA J. Appl. Math., 81 (2016), 165–198. https://doi.org/10.1093/imamat/hxv033 doi: 10.1093/imamat/hxv033
[37]	G. Q. Ren, B. Liu, Global dynamics for an attraction-repulsion chemotaxis model with logistic source, J. Diff. Equation, 268 (2022), 4320–4373. https://doi.org/10.1016/j.jde.2019.10.027 doi: 10.1016/j.jde.2019.10.027
[38]	S. J. Shi, Z. R. Liu, H. Y. Jin, Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source, Kinet. Relat. Mod., 10 (2017), 855–878. https://doi.org/10.3934/krm.2017034 doi: 10.3934/krm.2017034
[39]	M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math Phys., 69 (2018), 40–64. https://doi.org/10.1007/s00033-018-0935-8 doi: 10.1007/s00033-018-0935-8
[40]	Q. S. Zhang, Y. X. Li, An attraction-repulsion chemotaxis system with logistic source, Z. Angew. Math. Mech., 96 (2016), 570–584. https://doi.org/10.1002/zamm.201400311 doi: 10.1002/zamm.201400311
[41]	E. Nakaguchi, M. Efendiev, On a new dimension estimate of the global attractor for chemotaxis-growth systems, Osaka J. Math., 45 (2008), 273–281.
[42]	E. Nakaguchi, K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286–297. https://doi.org/10.1016/j.na.2010.08.044 doi: 10.1016/j.na.2010.08.044
[43]	E. Nakaguchi, K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013), 2627–2646. https://doi.org/10.3934/dcdsb.2013.18.2627 doi: 10.3934/dcdsb.2013.18.2627
[44]	M. Liu, Y. X. Li, Finite-time blowup in attraction-repulsion systems with nonlinear signal production, Nonlinear Anal. RWA., 61 (2021), 103305. https://doi.org/10.1016/j.nonrwa.2021.103305 doi: 10.1016/j.nonrwa.2021.103305
[45]	T. Black, Sublinear signal production in two-dimensional Keller-Segel-Stokes system, Nonlinear Anal. RWA., 31 (2016), 593–609. https://doi.org/10.1016/j.nonrwa.2016.03.008 doi: 10.1016/j.nonrwa.2016.03.008
[46]	D. M. Liu, Y. S. Tao, Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univser., 31 (2016), 379–388. https://doi.org/10.1007/s11766-016-3386-z doi: 10.1007/s11766-016-3386-z
[47]	T. Senba, T. Suzuki, Parabolic system of chemotaxis: blow-up in a infinite time, Methods Appl. Anal., 8 (2001), 349–367.
[48]	Y. S. Tao, M. Winkler, A chemotaxis-haptotaxis system with haptoattractant remodeling: boundedness enforced by mild saturation of signal production, Commun. Pure Appl. Anal., 18 (2019), 2047–2067. https://doi.org/10.3934/cpaa.2019092 doi: 10.3934/cpaa.2019092
[49]	O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Uralceva, Linear and quasilinear equations of parabolic type, in Translated form Russian by S. Smith. Translations of Mathematical Monographs, American Mathematical Society, (1968).
[50]	G. M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl., 148 (1987), 77–99. https://doi.org/10.1007/bf01774284 doi: 10.1007/bf01774284
[51]	N. D. Alikakos, $L^{p}$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differ. Equation, 4 (1978), 827–868. https://doi.org/10.1080/03605307908820113 doi: 10.1080/03605307908820113

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