# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems - B, 2013, 18 (3) : 821-845. doi: 10.3934/dcdsb.2013.18.821
##### 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.

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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.
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