May  2013, 18(3): 821-845. doi: 10.3934/dcdsb.2013.18.821

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,, United States

3. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received  March 2012 Revised  September 2012 Published  December 2012

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.
Citation: Youshan Tao, Lihe Wang, Zhi-An Wang. Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 821-845. doi: 10.3934/dcdsb.2013.18.821
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T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

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D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.- Verien., 105 (2003), 103. Google Scholar

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility,, J. Theor. Biol., 26 (1970), 399. Google Scholar

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E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis,, J. Theor. Biol., 26 (1971), 235. Google Scholar

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O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-linear Equations of Parabolic Type,", AMS, (1968). Google Scholar

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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. doi: 10.1137/S0036139995291106. Google Scholar

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D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis,, Math. Models Methods Appl. Sci., 21 (2011), 1631. doi: 10.1142/S0218202511005519. Google Scholar

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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. doi: 10.1137/110829453. Google Scholar

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T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (): 1522. doi: 10.1137/09075161X. Google Scholar

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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. doi: 10.1142/S0218202510004830. Google Scholar

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T. Li and Z.-A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,, J. Differential Equations, 250 (2011), 1310. doi: 10.1016/j.jde.2010.09.020. Google Scholar

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G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific, (1996). Google Scholar

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C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1998), 1. doi: 10.1016/0022-0396(88)90147-7. Google Scholar

[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. doi: 10.1080/17513758.2011.571722. Google Scholar

[27]

W.-M. Ni, Diffusion, cross-diffusion, and theri spike-layer steady states,, Notice of the AMS, 45 (1998), 9. Google Scholar

[28]

L. Nirenberg, An extended interpolation inequality,, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733. Google Scholar

[29]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441. Google Scholar

[30]

A. J. Perumpanani and H. M. Byrne, Extracellular matrix concentration exerts selection pressure on invasive cells,, Eur. J. Cancer, 35 (1999), 1274. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/S0036139995288976. Google Scholar

[33]

B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis,, Math. Methods Appl. Sci., 24 (2001), 405. doi: 10.1002/mma.212. Google Scholar

[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. doi: 10.1016/j.nonrwa.2011.07.006. Google Scholar

[35]

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

[36]

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

[37]

Z.-A. Wang and T. Hillen, Shock formation in a chemotaxis model,, Math. Methods. Appl. Sci., 31 (2008), 45. doi: 10.1002/mma.898. Google Scholar

[38]

Z.-A. Wang and K. Zhao, Global dynamics and diffusion limit of a repulsive chemotaxis model,, Comm. Pure and Appl. Anal., (). Google Scholar

[39]

M. Winkler, A critical exponent in a degenerate parabolic equation,, Math. Methods Appl. Sci., 25 (2002), 911. doi: 10.1002/mma.319. Google Scholar

[40]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Math. Methods Appl. Sci., 34 (2011), 176. doi: 10.1002/mma.1346. Google Scholar

[41]

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

[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. doi: 10.1017/S0308210500004649. Google Scholar

[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. doi: 10.1137/S0036141000337796. Google Scholar

[44]

Y. Yang, H. Chen, W. Liu and B. D. Sleeman, The solvability of some chemotaxis systems,, J. Diff. Eqn., 212 (2005), 432. doi: 10.1016/j.jde.2005.01.002. Google Scholar

[45]

M. Zhang and C. J. Zhu, Global existence of solutions to a hyperbolic-parabolic system,, Proc. Amer. Math. Soc., 135 (2007), 1017. doi: 10.1090/S0002-9939-06-08773-9. Google Scholar

show all references

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]

F. Almgren and L. Wang, Mathematical existence of crystal growth with Gibbs-Thomson curvature effects,, J. Geom. Anal., 10 (2000), 1. doi: 10.1007/BF02921806. Google Scholar

[3]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis,, Adv. Math. Sci. Appl., 9 (1999), 347. Google Scholar

[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. doi: 10.1007/s006050170032. Google Scholar

[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. doi: 10.1016/j.anihpc.2009.11.016. Google Scholar

[6]

T. Cieślak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady-states in a chemorepulsion system,, in, 81 (2008), 105. doi: 10.4064/bc81-0-7. Google Scholar

[7]

L. C. Evans, "Partial Differential Equations,", AMS, (1998). Google Scholar

[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. doi: 10.1137/S0036141001385046. Google Scholar

[9]

A. Friedman, "Partial Differential Equations,", Holt, (1969). Google Scholar

[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. doi: 10.1016/0022-1236(91)90136-S. Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1983). Google Scholar

[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. doi: 10.1016/S0252-9602(09)60059-X. Google Scholar

[13]

T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[14]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.- Verien., 105 (2003), 103. Google Scholar

[15]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility,, J. Theor. Biol., 26 (1970), 399. Google Scholar

[16]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis,, J. Theor. Biol., 26 (1971), 235. Google Scholar

[17]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-linear Equations of Parabolic Type,", AMS, (1968). Google Scholar

[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. doi: 10.1137/S0036139995291106. Google Scholar

[19]

D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis,, Math. Models Methods Appl. Sci., 21 (2011), 1631. doi: 10.1142/S0218202511005519. Google Scholar

[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. doi: 10.1137/110829453. Google Scholar

[21]

T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (): 1522. doi: 10.1137/09075161X. Google Scholar

[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. doi: 10.1142/S0218202510004830. Google Scholar

[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. doi: 10.1016/j.jde.2010.09.020. Google Scholar

[24]

G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific, (1996). Google Scholar

[25]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1998), 1. doi: 10.1016/0022-0396(88)90147-7. Google Scholar

[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. doi: 10.1080/17513758.2011.571722. Google Scholar

[27]

W.-M. Ni, Diffusion, cross-diffusion, and theri spike-layer steady states,, Notice of the AMS, 45 (1998), 9. Google Scholar

[28]

L. Nirenberg, An extended interpolation inequality,, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733. Google Scholar

[29]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441. Google Scholar

[30]

A. J. Perumpanani and H. M. Byrne, Extracellular matrix concentration exerts selection pressure on invasive cells,, Eur. J. Cancer, 35 (1999), 1274. Google Scholar

[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. Google Scholar

[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. doi: 10.1137/S0036139995288976. Google Scholar

[33]

B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis,, Math. Methods Appl. Sci., 24 (2001), 405. doi: 10.1002/mma.212. Google Scholar

[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. doi: 10.1016/j.nonrwa.2011.07.006. Google Scholar

[35]

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

[36]

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

[37]

Z.-A. Wang and T. Hillen, Shock formation in a chemotaxis model,, Math. Methods. Appl. Sci., 31 (2008), 45. doi: 10.1002/mma.898. Google Scholar

[38]

Z.-A. Wang and K. Zhao, Global dynamics and diffusion limit of a repulsive chemotaxis model,, Comm. Pure and Appl. Anal., (). Google Scholar

[39]

M. Winkler, A critical exponent in a degenerate parabolic equation,, Math. Methods Appl. Sci., 25 (2002), 911. doi: 10.1002/mma.319. Google Scholar

[40]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Math. Methods Appl. Sci., 34 (2011), 176. doi: 10.1002/mma.1346. Google Scholar

[41]

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

[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. doi: 10.1017/S0308210500004649. Google Scholar

[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. doi: 10.1137/S0036141000337796. Google Scholar

[44]

Y. Yang, H. Chen, W. Liu and B. D. Sleeman, The solvability of some chemotaxis systems,, J. Diff. Eqn., 212 (2005), 432. doi: 10.1016/j.jde.2005.01.002. Google Scholar

[45]

M. Zhang and C. J. Zhu, Global existence of solutions to a hyperbolic-parabolic system,, Proc. Amer. Math. Soc., 135 (2007), 1017. doi: 10.1090/S0002-9939-06-08773-9. Google Scholar

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