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

Xie Li; Zhaoyin Xiang

doi:10.3934/dcds.2015.35.3503

Article Contents

2015, Volume 35, Issue 8: 3503-3531. Doi: 10.3934/dcds.2015.35.3503

This issue Previous Article On the partitions with Sturmian-like refinements Next Article Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation

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

Xie Li^1, and
Zhaoyin Xiang^1,

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

Received: November 2014

Revised: December 2014

Published: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

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).

Keywords:

Global existence,
boundedness,
parabolic-elliptic Keller-Segel system,
parabolic-parabolic Keller-Segel systems.

Mathematics Subject Classification: Primary: 35K35, 92C17; Secondary: 35K59.

Citation:

Related Papers

Cited by

References

[1]	N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.doi: 10.1080/03605307908820113.
[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), 585-591.doi: 10.1016/j.crma.2013.07.027.
[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), 5215-5228.doi: 10.1016/j.na.2012.04.038.
[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), 2326-2330.doi: 10.1002/mma.2992.
[5]	T. Cieslak, Quasilinear nonuniformly parabolic system modelling chemotaxis, J. Math. Anal. Appl., 326 (2007), 1410-1426.doi: 10.1016/j.jmaa.2006.03.080.
[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), 437-446.doi: 10.1016/j.anihpc.2009.11.016.
[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), 5832-5851.doi: 10.1016/j.jde.2012.01.045.
[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), 135-146.doi: 10.1007/s10440-013-9832-5.
[9]	T. Cieslak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.doi: 10.1088/0951-7715/21/5/009.
[10]	A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[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, Res. Inst. Math. Sci. (RIMS), Kyoto, 26 (2011), 159-175.
[12]	D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008.
[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), 633-683.
[14]	T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.doi: 10.1006/aama.2001.0721.
[15]	T. Hillen and K. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
[16]	T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.doi: 10.1007/s00285-008-0201-3.
[17]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.doi: 10.1016/j.jde.2004.10.022.
[18]	D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.doi: 10.1017/S0956792501004363.
[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), 745-760.doi: 10.1002/mma.2622.
[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), 2993-3010.doi: 10.1016/j.jde.2014.01.028.
[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), 819-824.doi: 10.1090/S0002-9947-1992-1046835-6.
[22]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.doi: 10.1016/0022-5193(70)90092-5.
[23]	N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. I. H. Poincaré AN, 31 (2014), 851-875.doi: 10.1016/j.anihpc.2013.07.007.
[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), 1634-1642.doi: 10.1016/j.nonrwa.2012.10.022.
[25]	T. Nagai, Blow-up of radially symmetric solutions of a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
[26]	E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297.doi: 10.1016/j.na.2010.08.044.
[27]	V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theoret. Biol., 42 (1973), 63-105.doi: 10.1016/0022-5193(73)90149-5.
[28]	L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733-737.
[29]	T. Senba, Blowup behavior of radial solutions to Jager-Luckhaus system in high dimensional domains, Funkcialaj Ekvacioj, 48 (2005), 247-271.doi: 10.1619/fesi.48.247.
[30]	T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.
[31]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.doi: 10.1016/j.jde.2011.08.019.
[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), 2520-2543.doi: 10.1016/j.jde.2011.07.010.
[33]	Y. Tao and Z.A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.doi: 10.1142/S0218202512500443.
[34]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.doi: 10.1080/03605300701319003.
[35]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Appl. Math. Sci., 68, Springer-Verlag, NewYork, 1997.
[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), 789-802.doi: 10.3934/dcds.2014.34.789.
[37]	L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.doi: 10.1016/j.jde.2013.12.007.
[38]	Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13pp.doi: 10.1063/1.2766864.
[39]	Z. A. Wang, On chemotaxis models with cell population interactions, Math. Model. Nat. Phenom., 5 (2010), 173-190.doi: 10.1051/mmnp/20105311.
[40]	Z. A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297.doi: 10.1088/0951-7715/24/12/001.
[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), 3502-3525.doi: 10.1137/110853972.
[42]	M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.doi: 10.1080/03605300903473426.
[43]	M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.doi: 10.1016/j.jmaa.2008.07.071.
[44]	M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.doi: 10.1016/j.matpur.2013.01.020.
[45]	M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.doi: 10.1002/mma.1146.
[46]	M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.doi: 10.1016/j.jde.2010.02.008.
[47]	M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.doi: 10.1016/j.na.2009.07.045.
[48]	D. Wrzosek, Global attractor for a chemotaxis model with prevention of over-crowding, Nonlinear Anal., 59 (2004), 1293-1310.doi: 10.1016/j.na.2004.08.015.