December  2014, 34(12): 5165-5179. doi: 10.3934/dcds.2014.34.5165

On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical

1. 

Department of Mathematics, Yonsei University, Seoul, South Korea

Received  August 2013 Revised  March 2014 Published  June 2014

We consider a chemotactic system with a logarithmic sensitivity and a non-diffusing chemical. We establish local regular solutions in time and give some characterizations on parameters and initial data for global solutions and blow-up in a finite time. We also prove that there does not exist finite time self-similar solution of the backward type.
Citation: Jaewook Ahn, Kyungkeun Kang. On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5165-5179. doi: 10.3934/dcds.2014.34.5165
References:
[1]

H. Chen, W. Liu and Y. Yang, On existence of global solutions and blow-up to a system of reaction-diffusion equations modelling chemotaxis,, SIAM J. Math. Anal., 33 (2001), 763.  doi: 10.1137/S0036141000337796.  Google Scholar

[2]

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

[3]

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

[4]

A. Friedman and I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks,, J. Math. Anal. Appl., 272 (2002), 138.  doi: 10.1016/S0022-247X(02)00147-6.  Google Scholar

[5]

D. Henry, Geometric Theory of Semilinear Parabolic Equation,, Springer-Verlag, (1981).   Google Scholar

[6]

K. Kang, A. Stevens and J. J. L. Velázquez, Qualitative behavior of a Keller-Segel model with non-diffusive memory,, J. Comm. Part. Diff. Eqs., 35 (2010), 245.  doi: 10.1080/03605300903473400.  Google Scholar

[7]

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

[8]

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

[9]

H. Kozono, Y. Sugiyama and R. Takada, Non-existence of finite-time self-similar solutions of the Keller-Segel system in the scaling invariant class,, J. Math. Anal. Appl., 365 (2010), 60.  doi: 10.1016/j.jmaa.2009.09.063.  Google Scholar

[10]

O. A. Ladyženskja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Transl. Math. Monogr.,, AMS., (1967).   Google Scholar

[11]

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

[12]

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

[13]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co. Inc., (1996).  doi: 10.1142/3302.  Google Scholar

[14]

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

[15]

B. Perthame and A. Vasseur, Regularization in Keller Segel type systems and the De Giorgi method,, J. Comm. Math. Sci., 10 (2012), 463.  doi: 10.4310/CMS.2012.v10.n2.a2.  Google Scholar

[16]

A. Stevens, Trail following and aggregation of myxobacteria,, J. Biol. Systems, 3 (1995), 1059.  doi: 10.1142/S0218339095000952.  Google Scholar

[17]

A. Stevens and J. J. L. Velázquez, Asymptotic analysis of a chemotaxis system with non-diffusive memory,, preprint., ().   Google Scholar

[18]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems,, J. Diff. Int. Eqns., 19 (2006), 841.   Google Scholar

[19]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Diff. Eqs., 248 (2010), 2889.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

show all references

References:
[1]

H. Chen, W. Liu and Y. Yang, On existence of global solutions and blow-up to a system of reaction-diffusion equations modelling chemotaxis,, SIAM J. Math. Anal., 33 (2001), 763.  doi: 10.1137/S0036141000337796.  Google Scholar

[2]

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

[3]

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

[4]

A. Friedman and I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks,, J. Math. Anal. Appl., 272 (2002), 138.  doi: 10.1016/S0022-247X(02)00147-6.  Google Scholar

[5]

D. Henry, Geometric Theory of Semilinear Parabolic Equation,, Springer-Verlag, (1981).   Google Scholar

[6]

K. Kang, A. Stevens and J. J. L. Velázquez, Qualitative behavior of a Keller-Segel model with non-diffusive memory,, J. Comm. Part. Diff. Eqs., 35 (2010), 245.  doi: 10.1080/03605300903473400.  Google Scholar

[7]

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

[8]

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

[9]

H. Kozono, Y. Sugiyama and R. Takada, Non-existence of finite-time self-similar solutions of the Keller-Segel system in the scaling invariant class,, J. Math. Anal. Appl., 365 (2010), 60.  doi: 10.1016/j.jmaa.2009.09.063.  Google Scholar

[10]

O. A. Ladyženskja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Transl. Math. Monogr.,, AMS., (1967).   Google Scholar

[11]

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

[12]

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

[13]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co. Inc., (1996).  doi: 10.1142/3302.  Google Scholar

[14]

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

[15]

B. Perthame and A. Vasseur, Regularization in Keller Segel type systems and the De Giorgi method,, J. Comm. Math. Sci., 10 (2012), 463.  doi: 10.4310/CMS.2012.v10.n2.a2.  Google Scholar

[16]

A. Stevens, Trail following and aggregation of myxobacteria,, J. Biol. Systems, 3 (1995), 1059.  doi: 10.1142/S0218339095000952.  Google Scholar

[17]

A. Stevens and J. J. L. Velázquez, Asymptotic analysis of a chemotaxis system with non-diffusive memory,, preprint., ().   Google Scholar

[18]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems,, J. Diff. Int. Eqns., 19 (2006), 841.   Google Scholar

[19]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Diff. Eqs., 248 (2010), 2889.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

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