January  2019, 24(1): 183-195. doi: 10.3934/dcdsb.2018093

Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces

1. 

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

2. 

Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA

* Corresponding author: Tong Li

Received  December 2016 Revised  September 2017 Published  March 2018

Fund Project: The first author is supported by NSFC (No. 10931001)

This paper is devoted to the global analysis for the two-dimensional parabolic-parabolic Keller-Segel system in the whole space. By well balanced arguments of the $L^1$ and $L^∞$ spaces, we first prove global well-posedness of the system in $L^1× L^∞$ which partially answers the question posted by Kozono et al in [19]. For the case $μ_0>0$, we make full use of the linear parts of the system to get the improved long time decay property. Moreover, by using the new formulation involving all linear parts, introducing the logarithmic-weight in time to modify the other endpoint space $L^∞× L^∞$, and carefully decomposing time into several pieces, we are able to establish the global well-posedness and large time behavior of the system in $L^∞_{ln}× L^∞$.

Citation: Chao Deng, Tong Li. Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 183-195. doi: 10.3934/dcdsb.2018093
References:
[1]

A. BlanchetJ. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Diff. Eqns., 44 (2006), 1-32. Google Scholar

[2]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $R^2$, Commun. Math. Sci., 6 (2008), 417-447. doi: 10.4310/CMS.2008.v6.n2.a8. Google Scholar

[3]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9. Google Scholar

[4]

S. Childress and J. K. Percus, Chemotactic collapse in two dimensions, Lecture Notes in Biomathematics, 55, Springer, Berlin-Heidelberg-New York, 61-66,1984. Google Scholar

[5]

L. CorriasB. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris, Ser. Ⅰ., 336 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0. Google Scholar

[6]

L. CorriasB. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x. Google Scholar

[7]

J. I. DiazT. Nagai and J. M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^n$, J. Differential Equations, 145 (1998), 156-183. doi: 10.1006/jdeq.1997.3389. Google Scholar

[8]

C. Deng and T. Li, Well-posedness of the 3D Parabolic-hyperbolic Keller-Segel System in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332. doi: 10.1016/j.jde.2014.05.014. Google Scholar

[9]

M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004.Google Scholar

[10]

Y. Guo and H. J. Hwang, Pattern formation (Ⅰ): The Keller-Segel model, J. Differential Equations, 249 (2010), 1519-1530. doi: 10.1016/j.jde.2010.07.025. Google Scholar

[11]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences Ⅰ., Jahresber. Dutsch. Math. Ver., 105 (2003), 103-165. Google Scholar

[12]

Y. Kagei and Y. Maekawa, On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis, J. Differential Equations, 253 (2012), 2951-2992. doi: 10.1016/j.jde.2012.08.028. Google Scholar

[13]

T. Kato, Strong ${L}^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182. Google Scholar

[14]

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. Google Scholar

[15]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[16]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8. Google Scholar

[17]

H. Kozono and Y. Sugiyama, Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces, J. Differential Equations, 247 (2009), 1-32. doi: 10.1016/j.jde.2009.03.027. Google Scholar

[18]

H. Kozono and Y. Sugiyama, Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}$ and its application to self-similar solutions, Indiana Univ. Math. J., 57 (2008), 1467-1500. doi: 10.1512/iumj.2008.57.3316. Google Scholar

[19]

H. KozonoY. Sugiyama and T. Wachi, Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space, J. Differential Equations, 252 (2012), 1213-1228. doi: 10.1016/j.jde.2011.08.025. Google Scholar

[20]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Research Notes in Mathematics, Chapman & Hall/CRC, 2002. Google Scholar

[21]

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. Google Scholar

[22]

D. LiT. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Model. Meth. Appl. Sci., 21 (2011), 1631-1650. doi: 10.1142/S0218202511005519. Google Scholar

[23]

T. LiR. 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. Google Scholar

[24]

T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541. Google Scholar

[25]

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. Google Scholar

[26]

C. S. LinW. 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. Google Scholar

[27]

T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184. doi: 10.1007/BF00160334. Google Scholar

[28]

H. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976. Google Scholar

[29]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar

[30]

B. D. SleemanM. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117. Google Scholar

[31]

Z. A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Meth. Appl. Sci., 31 (2008), 45-70. doi: 10.1002/mma.898. Google Scholar

[32]

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. Google Scholar

[33]

Y. YangH. ChenW. Liu and B. D. Sleeman, The solvability of some chemotaxis systems, J. Differential Equations, 212 (2005), 432-451. doi: 10.1016/j.jde.2005.01.002. Google Scholar

show all references

References:
[1]

A. BlanchetJ. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Diff. Eqns., 44 (2006), 1-32. Google Scholar

[2]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $R^2$, Commun. Math. Sci., 6 (2008), 417-447. doi: 10.4310/CMS.2008.v6.n2.a8. Google Scholar

[3]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9. Google Scholar

[4]

S. Childress and J. K. Percus, Chemotactic collapse in two dimensions, Lecture Notes in Biomathematics, 55, Springer, Berlin-Heidelberg-New York, 61-66,1984. Google Scholar

[5]

L. CorriasB. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris, Ser. Ⅰ., 336 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0. Google Scholar

[6]

L. CorriasB. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x. Google Scholar

[7]

J. I. DiazT. Nagai and J. M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^n$, J. Differential Equations, 145 (1998), 156-183. doi: 10.1006/jdeq.1997.3389. Google Scholar

[8]

C. Deng and T. Li, Well-posedness of the 3D Parabolic-hyperbolic Keller-Segel System in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332. doi: 10.1016/j.jde.2014.05.014. Google Scholar

[9]

M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004.Google Scholar

[10]

Y. Guo and H. J. Hwang, Pattern formation (Ⅰ): The Keller-Segel model, J. Differential Equations, 249 (2010), 1519-1530. doi: 10.1016/j.jde.2010.07.025. Google Scholar

[11]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences Ⅰ., Jahresber. Dutsch. Math. Ver., 105 (2003), 103-165. Google Scholar

[12]

Y. Kagei and Y. Maekawa, On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis, J. Differential Equations, 253 (2012), 2951-2992. doi: 10.1016/j.jde.2012.08.028. Google Scholar

[13]

T. Kato, Strong ${L}^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182. Google Scholar

[14]

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. Google Scholar

[15]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[16]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8. Google Scholar

[17]

H. Kozono and Y. Sugiyama, Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces, J. Differential Equations, 247 (2009), 1-32. doi: 10.1016/j.jde.2009.03.027. Google Scholar

[18]

H. Kozono and Y. Sugiyama, Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}$ and its application to self-similar solutions, Indiana Univ. Math. J., 57 (2008), 1467-1500. doi: 10.1512/iumj.2008.57.3316. Google Scholar

[19]

H. KozonoY. Sugiyama and T. Wachi, Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space, J. Differential Equations, 252 (2012), 1213-1228. doi: 10.1016/j.jde.2011.08.025. Google Scholar

[20]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Research Notes in Mathematics, Chapman & Hall/CRC, 2002. Google Scholar

[21]

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. Google Scholar

[22]

D. LiT. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Model. Meth. Appl. Sci., 21 (2011), 1631-1650. doi: 10.1142/S0218202511005519. Google Scholar

[23]

T. LiR. 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. Google Scholar

[24]

T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541. Google Scholar

[25]

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. Google Scholar

[26]

C. S. LinW. 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. Google Scholar

[27]

T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184. doi: 10.1007/BF00160334. Google Scholar

[28]

H. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976. Google Scholar

[29]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar

[30]

B. D. SleemanM. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117. Google Scholar

[31]

Z. A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Meth. Appl. Sci., 31 (2008), 45-70. doi: 10.1002/mma.898. Google Scholar

[32]

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. Google Scholar

[33]

Y. YangH. ChenW. Liu and B. D. Sleeman, The solvability of some chemotaxis systems, J. Differential Equations, 212 (2005), 432-451. doi: 10.1016/j.jde.2005.01.002. Google Scholar

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