Large-time behavior of a  parabolic-parabolic chemotaxis modelwith logarithmicsensitivity in one dimension

Youshan Tao; Lihe Wang; Zhi-An Wang

doi:10.3934/dcdsb.2013.18.821

Article Contents

2013, Volume 18, Issue 3: 821-845. Doi: 10.3934/dcdsb.2013.18.821

This issue Previous Article The long time behavior of a spectral collocation method for delay differential equations of pantograph type Next Article

Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension

1.
Department of Applied Mathematics, Dong Hua University, Shanghai 200051
2.
Department of Mathematics, 15 MLH, The University of Iowa, Iowa City, IA 52242-1419,
3.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received: March 2012

Revised: September 2012

Published: May 2013

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Abstract

This paper deals with the chemotaxis system $$ \left\{ \begin{array}{ll} u_t ={D} u_{xx}-\chi [u(\ln v)_x]_x, & x\in (0, 1), \ t>0,\\ v_t =\varepsilon v_{xx} +uv-\mu v, & x\in (0, 1), \ t>0, \end{array} \right. $$ under Neumann boundary condition, where $\chi<0$, $D>0$, $\varepsilon>0$ and $\mu>0$ are constants.
It is shown that for any sufficiently smooth initial data $(u_0, v_0)$ fulfilling $u_0\ge 0$, $u_0 \not\equiv 0$ and $v_0>0$, the system possesses a unique global smooth solution that enjoys exponential convergence properties in $L^\infty(\Omega)$ as time goes to infinity, which depend on the sign of $\mu-\bar{u}_0$, where $\bar{u}_0 :=\int_0^1 u_0 dx$. Moreover, we prove that the constant pair $(\mu, (\frac{\mu}{\lambda})^{\frac{D}{\chi}})$ (where $\lambda>0$ is an arbitrary constant) is the only positive stationary solution. The biological implications of our results will be given in the paper.

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Mathematics Subject Classification: Primary: 35A01, 35B40, 35B44, 35K57; Secondary: 35Q92,92C17.

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References

[1]	N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.doi: 10.1080/03605307908820113.
[2]	F. Almgren and L. Wang, Mathematical existence of crystal growth with Gibbs-Thomson curvature effects, J. Geom. Anal., 10 (2000), 1-100.doi: 10.1007/BF02921806.
[3]	P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359.
[4]	J. A. Carrillo, A. Jüngle, P. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.doi: 10.1007/s006050170032.
[5]	T. Cieślak and P. Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 437-446.doi: 10.1016/j.anihpc.2009.11.016.
[6]	T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system, in "Parabolic and Navier-Stokes Equations. Part 1," Banach Center Publ., 81, Polish Acad. Sci., Warsaw, (2008), 105-117.doi: 10.4064/bc81-0-7.
[7]	L. C. Evans, "Partial Differential Equations," AMS, Providence, 1998.
[8]	M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.doi: 10.1137/S0036141001385046.
[9]	A. Friedman, "Partial Differential Equations," Holt, Rinehart & Winston, New York, 1969.
[10]	Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Analysis, 102 (1991), 72-94.doi: 10.1016/0022-1236(91)90136-S.
[11]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, New York, 1983.
[12]	J. Guo, J. X. Xiao, H. J. Zhao and C. J. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 629-641.doi: 10.1016/S0252-9602(09)60059-X.
[13]	T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.doi: 10.1007/s00285-008-0201-3.
[14]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.- Verien., 105 (2003), 103-165.
[15]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415.
[16]	E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 26 (1971), 235-248.
[17]	O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-linear Equations of Parabolic Type," AMS, Providence, 1968.
[18]	H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.doi: 10.1137/S0036139995291106.
[19]	D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 1631-1650.doi: 10.1142/S0218202511005519.
[20]	T. Li, R. H. Pan and K. Zhao, Global dynamics of a chemotaxis model on bounded domains with large data, SIAM J. Appl. Math., 72 (2012), 417-443.doi: 10.1137/110829453.
[21]	T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009/10), 1522-1541. doi: 10.1137/09075161X.
[22]	T. Li and Z.-A. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967-1998.doi: 10.1142/S0218202510004830.
[23]	T. Li and Z.-A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.doi: 10.1016/j.jde.2010.09.020.
[24]	G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, Singapore, 1996.
[25]	C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1998), 1-27.doi: 10.1016/0022-0396(88)90147-7.
[26]	J. Liu and Z.-A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.doi: 10.1080/17513758.2011.571722.
[27]	W.-M. Ni, Diffusion, cross-diffusion, and theri spike-layer steady states, Notice of the AMS, 45 (1998), 9-18.
[28]	L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737.
[29]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
[30]	A. J. Perumpanani and H. M. Byrne, Extracellular matrix concentration exerts selection pressure on invasive cells, Eur. J. Cancer, 35 (1999), 1274-1280.
[31]	A. J. Perumpanani, D. L. Simmons, A. J. H. Gearing, K. M. Miller, G. Ward, J. Norbury, M. Schneemann and J. A. Sherratt, Extracellular matrix-mediated chemotaxis can impede cell migration, Proc. R. Soc. Lond. B, 265 (1998), 2347-2352.
[32]	H. G. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.doi: 10.1137/S0036139995288976.
[33]	B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis, Math. Methods Appl. Sci., 24 (2001), 405-426.doi: 10.1002/mma.212.
[34]	Ch. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Analysis: Real World Applications, 12 (2011), 3727-3740.doi: 10.1016/j.nonrwa.2011.07.006.
[35]	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.
[36]	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.
[37]	Z.-A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods. Appl. Sci., 31 (2008), 45-70.doi: 10.1002/mma.898.
[38]	Z.-A. Wang and K. Zhao, Global dynamics and diffusion limit of a repulsive chemotaxis model, Comm. Pure and Appl. Anal., to appear.
[39]	M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.doi: 10.1002/mma.319.
[40]	M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.doi: 10.1002/mma.1346.
[41]	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.
[42]	D. Wrzosek, Long-time behaviour of solutions to a chemotaxis model with volume-filling effect, Proceedings of the Royal Society of Edinburgy A, 136 (2006), 431-444.doi: 10.1017/S0308210500004649.
[43]	Y. Yang, H. Chen and W. Liu, On existence of global solutions and blow-up to a system of the reaction-diffusion equations modelling chemotaxis, SIAM J. Math. Anal., 33 (2001), 763-785.doi: 10.1137/S0036141000337796.
[44]	Y. Yang, H. Chen, W. Liu and B. D. Sleeman, The solvability of some chemotaxis systems, J. Diff. Eqn., 212 (2005), 432-451.doi: 10.1016/j.jde.2005.01.002.
[45]	M. Zhang and C. J. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017-1027.doi: 10.1090/S0002-9939-06-08773-9.