American Institute of Mathematical Sciences

August  2015, 35(8): 3503-3531. doi: 10.3934/dcds.2015.35.3503

Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source

 1 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731

Received  November 2014 Revised  December 2014 Published  February 2015

In this paper, we investigate the quasilinear Keller-Segel equations (q-K-S): $\left\{ \begin{split} &n_t=\nabla\cdot\big(D(n)\nabla n\big)-\nabla\cdot\big(\chi(n)\nabla c\big)+\mathcal{R}(n), \qquad x\in\Omega,\,t>0,\\ &\varrho c_t=\Delta c-c+n, \qquad x\in\Omega,\,t>0, \end{split} \right.$ under homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^N$. For both $\varrho=0$ (parabolic-elliptic case) and $\varrho>0$ (parabolic-parabolic case), we will show the global-in-time existence and uniform-in-time boundedness of solutions to equations (q-K-S) with both non-degenerate and degenerate diffusions on the non-convex domain $\Omega$, which provide a supplement to the dichotomy boundedness vs. blow-up in parabolic-elliptic/parabolic-parabolic chemotaxis equations with degenerate diffusion, nonlinear sensitivity and logistic source. In particular, we improve the recent results obtained by Wang-Li-Mu (2014, Disc. Cont. Dyn. Syst.) and Wang-Mu-Zheng (2014, J. Differential Equations).
Citation: Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503
References:
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Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations,, in Harmonic Analysis and Nonlinear Partial Differential Equations, 26 (2011), 159. Google Scholar [12] D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations,, European Mathematical Society, (2008). Google Scholar [13] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633. Google Scholar [14] T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding,, Adv. Appl. Math., 26 (2001), 280. doi: 10.1006/aama.2001.0721. Google Scholar [15] T. Hillen and K. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501. Google Scholar [16] T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar [17] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar [18] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math., 12 (2001), 159. doi: 10.1017/S0956792501004363. Google Scholar [19] S. Ishida, T. Ono and T. Yokota, Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, Math. Methods Appl. Sci., 36 (2013), 745. doi: 10.1002/mma.2622. Google Scholar [20] S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains,, J. Differential Equations, 256 (2014), 2993. doi: 10.1016/j.jde.2014.01.028. Google Scholar [21] W. Jager and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar [22] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [23] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system,, Ann. I. H. Poincaré AN, 31 (2014), 851. doi: 10.1016/j.anihpc.2013.07.007. Google Scholar [24] C. Mu, L. Wang, P. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system,, Nonlinear Anal.-Real World Appl., 14 (2013), 1634. doi: 10.1016/j.nonrwa.2012.10.022. Google Scholar [25] T. Nagai, Blow-up of radially symmetric solutions of a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581. Google Scholar [26] E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation,, Nonlinear Anal., 74 (2011), 286. doi: 10.1016/j.na.2010.08.044. Google Scholar [27] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, J. Theoret. Biol., 42 (1973), 63. doi: 10.1016/0022-5193(73)90149-5. Google Scholar [28] L. Nirenberg, An extended interpolation inequality,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733. Google Scholar [29] T. Senba, Blowup behavior of radial solutions to Jager-Luckhaus system in high dimensional domains,, Funkcialaj Ekvacioj, 48 (2005), 247. doi: 10.1619/fesi.48.247. Google Scholar [30] T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time,, Methods Appl. Anal., 8 (2001), 349. Google Scholar [31] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692. doi: 10.1016/j.jde.2011.08.019. Google Scholar [32] Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant,, J. Differential Equations, 252 (2012), 2520. doi: 10.1016/j.jde.2011.07.010. Google Scholar [33] Y. Tao and Z.A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1. doi: 10.1142/S0218202512500443. Google Scholar [34] J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003. Google Scholar [35] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, 68 (1997). Google Scholar [36] L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source,, Discrete and Continuous Dynamical Systems, 34 (2014), 789. doi: 10.3934/dcds.2014.34.789. Google Scholar [37] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source,, J. Differential Equations, 256 (2014), 1847. doi: 10.1016/j.jde.2013.12.007. Google Scholar [38] Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model,, Chaos, 17 (2007). doi: 10.1063/1.2766864. Google Scholar [39] Z. A. Wang, On chemotaxis models with cell population interactions,, Math. Model. Nat. Phenom., 5 (2010), 173. doi: 10.1051/mmnp/20105311. Google Scholar [40] Z. A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect,, Nonlinearity, 24 (2011), 3279. doi: 10.1088/0951-7715/24/12/001. Google Scholar [41] Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion,, SIAM J. Math. Anal., 44 (2012), 3502. doi: 10.1137/110853972. Google Scholar [42] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Comm. Partial Differential Equations, 35 (2010), 1516. doi: 10.1080/03605300903473426. Google Scholar [43] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties,, J. Math. Anal. Appl., 348 (2008), 708. doi: 10.1016/j.jmaa.2008.07.071. Google Scholar [44] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748. doi: 10.1016/j.matpur.2013.01.020. Google Scholar [45] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12. doi: 10.1002/mma.1146. Google Scholar [46] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [47] M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect,, Nonlinear Anal., 72 (2010), 1044. doi: 10.1016/j.na.2009.07.045. Google Scholar [48] D. Wrzosek, Global attractor for a chemotaxis model with prevention of over-crowding,, Nonlinear Anal., 59 (2004), 1293. doi: 10.1016/j.na.2004.08.015. Google Scholar

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References:
 [1] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827. doi: 10.1080/03605307908820113. Google Scholar [2] K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logistic source,, C. R. Acad. Sci. Paris, 351 (2013), 585. doi: 10.1016/j.crma.2013.07.027. Google Scholar [3] J. Burczak, T. Cieslak and C. Morales-Rodrigo, Global existence vs. blow-up in a fully parabolic quasilinear 1D Keller-Segel system,, Nonlinear Anal., 75 (2012), 5215. doi: 10.1016/j.na.2012.04.038. Google Scholar [4] X. Cao and S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source,, Math. Methods Appl. Sci., 37 (2014), 2326. doi: 10.1002/mma.2992. Google Scholar [5] T. Cieslak, Quasilinear nonuniformly parabolic system modelling chemotaxis,, J. Math. Anal. Appl., 326 (2007), 1410. doi: 10.1016/j.jmaa.2006.03.080. Google Scholar [6] T. Cieslak and P. Laurencot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. I. H. Poincaré AN, 27 (2010), 437. doi: 10.1016/j.anihpc.2009.11.016. Google Scholar [7] T. Cieslak and C. Stinner, Finite-time blow up and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, J. Differential Equations, 252 (2012), 5832. doi: 10.1016/j.jde.2012.01.045. Google Scholar [8] T. Cieslak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2,, Acta Appl. Math., 129 (2014), 135. doi: 10.1007/s10440-013-9832-5. Google Scholar [9] T. Cieslak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, Nonlinearity, 21 (2008), 1057. doi: 10.1088/0951-7715/21/5/009. Google Scholar [10] A. Friedman, Partial Differential Equations,, Holt, (1969). Google Scholar [11] H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations,, in Harmonic Analysis and Nonlinear Partial Differential Equations, 26 (2011), 159. Google Scholar [12] D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations,, European Mathematical Society, (2008). Google Scholar [13] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633. Google Scholar [14] T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding,, Adv. Appl. Math., 26 (2001), 280. doi: 10.1006/aama.2001.0721. Google Scholar [15] T. Hillen and K. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501. Google Scholar [16] T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar [17] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar [18] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math., 12 (2001), 159. doi: 10.1017/S0956792501004363. Google Scholar [19] S. Ishida, T. Ono and T. Yokota, Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, Math. Methods Appl. Sci., 36 (2013), 745. doi: 10.1002/mma.2622. Google Scholar [20] S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains,, J. Differential Equations, 256 (2014), 2993. doi: 10.1016/j.jde.2014.01.028. Google Scholar [21] W. Jager and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar [22] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [23] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system,, Ann. I. H. Poincaré AN, 31 (2014), 851. doi: 10.1016/j.anihpc.2013.07.007. Google Scholar [24] C. Mu, L. Wang, P. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system,, Nonlinear Anal.-Real World Appl., 14 (2013), 1634. doi: 10.1016/j.nonrwa.2012.10.022. Google Scholar [25] T. Nagai, Blow-up of radially symmetric solutions of a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581. Google Scholar [26] E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation,, Nonlinear Anal., 74 (2011), 286. doi: 10.1016/j.na.2010.08.044. Google Scholar [27] V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, J. Theoret. Biol., 42 (1973), 63. doi: 10.1016/0022-5193(73)90149-5. Google Scholar [28] L. Nirenberg, An extended interpolation inequality,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733. Google Scholar [29] T. Senba, Blowup behavior of radial solutions to Jager-Luckhaus system in high dimensional domains,, Funkcialaj Ekvacioj, 48 (2005), 247. doi: 10.1619/fesi.48.247. Google Scholar [30] T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time,, Methods Appl. Anal., 8 (2001), 349. Google Scholar [31] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692. doi: 10.1016/j.jde.2011.08.019. Google Scholar [32] Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant,, J. Differential Equations, 252 (2012), 2520. doi: 10.1016/j.jde.2011.07.010. Google Scholar [33] Y. Tao and Z.A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1. doi: 10.1142/S0218202512500443. Google Scholar [34] J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003. Google Scholar [35] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, 68 (1997). Google Scholar [36] L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source,, Discrete and Continuous Dynamical Systems, 34 (2014), 789. doi: 10.3934/dcds.2014.34.789. Google Scholar [37] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source,, J. Differential Equations, 256 (2014), 1847. doi: 10.1016/j.jde.2013.12.007. Google Scholar [38] Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model,, Chaos, 17 (2007). doi: 10.1063/1.2766864. Google Scholar [39] Z. A. Wang, On chemotaxis models with cell population interactions,, Math. Model. Nat. Phenom., 5 (2010), 173. doi: 10.1051/mmnp/20105311. Google Scholar [40] Z. A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect,, Nonlinearity, 24 (2011), 3279. doi: 10.1088/0951-7715/24/12/001. Google Scholar [41] Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion,, SIAM J. Math. Anal., 44 (2012), 3502. doi: 10.1137/110853972. Google Scholar [42] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Comm. Partial Differential Equations, 35 (2010), 1516. doi: 10.1080/03605300903473426. Google Scholar [43] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties,, J. Math. Anal. Appl., 348 (2008), 708. doi: 10.1016/j.jmaa.2008.07.071. Google Scholar [44] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748. doi: 10.1016/j.matpur.2013.01.020. Google Scholar [45] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12. doi: 10.1002/mma.1146. Google Scholar [46] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [47] M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect,, Nonlinear Anal., 72 (2010), 1044. doi: 10.1016/j.na.2009.07.045. Google Scholar [48] D. Wrzosek, Global attractor for a chemotaxis model with prevention of over-crowding,, Nonlinear Anal., 59 (2004), 1293. doi: 10.1016/j.na.2004.08.015. Google Scholar
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