# American Institute of Mathematical Sciences

December  2015, 8(4): 667-684. doi: 10.3934/krm.2015.8.667

## A free boundary problem for a class of parabolic type chemotaxis model

 1 School of Mathematics and Statistics, Wuhan University, Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China, China, China

Received  February 2015 Revised  May 2015 Published  July 2015

In this paper, we study a free boundary problem for a class of parabolic type chemotaxis model in high dimensional symmetry domain $\Omega$. By using the contraction mapping principle and operator semigroup approach, we establish the existence of the solution for such kind of chemotaxis system in the domain $\Omega$ with free boundary condition.
Citation: Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic type chemotaxis model. Kinetic and Related Models, 2015, 8 (4) : 667-684. doi: 10.3934/krm.2015.8.667
##### References:
 [1] H. Chen and S. H. Wu, On existence of solutions for some hyperbolic-parabolic-type chemotaxis systems, IMA Journal of Applied Mathematics, 72 (2007), 331-347. doi: 10.1093/imamat/hxm008. [2] H. Chen and S. H. Wu, The free boundary problem in biological phenomena, Journal of Partial Differential Equations, 20 (2007), 155-168. [3] H. Chen and S. H. Wu, Hyperbolic-parabolic chemotaxis system with nonlinear product terms, Journal of Partial Differential Equations, 21 (2008), 45-58. [4] H. Chen and S. H. Wu, Nonlinear hyperbolic-parabolic system modeling some biological phenomena, Journal of Partial Differential Equations, 24 (2011), 1-14. [5] H. Chen and S. H. Wu, The moving the moving boundary problem in a chemotaxis model, Communications on Pure and Applied Analysis, 11 (2012), 735-746. doi: 10.3934/cpaa.2012.11.735. [6] H. Chen and X. H. Zhong, Norm behaviour of solutions to a parabolic-elliptic system modelling chemotaxis in a domain of $\mathbbR^3$, Mathematical Methods in the Applied Sciences, 27 (2004), 991-1006. doi: 10.1002/mma.479. [7] H. Chen and X. H. Zhong, Global existence and blow-up for the solutions to nonlinear parabolic-elliptic system modelling chemotaxis, IMA Journal of Applied Mathematics, 70 (2005), 221-240. doi: 10.1093/imamat/hxh032. [8] H. Chen and X. H. Zhong, Existence and stability of steady solutions to nonlinear parabolic-elliptic systems modelling chemotaxis, Mathematische Nachrichten, 279 (2006), 1441-1447. doi: 10.1002/mana.200310430. [9] A. Friedman, Free boundary problems in science and technology, Notices of the American Mathematical Society, 47 (2000), 854-861. [10] H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Mathematische Nachrichten, 195 (1998), 77-114. doi: 10.1002/mana.19981950106. [11] T. Hillen and K. J. Painter, A user's guide to pde models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [12] D. Horstmann, From 1970 until present: The keller-segel model in chemotaxis and its consequences I, Jahresbericht der Deutschen Mathematiker Vereinigung, 105 (2003), 103-165. [13] D. Horstmann and G. F. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European Journal of Applied Mathematics, 12 (2001), 159-177. doi: 10.1017/S0956792501004363. [14] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [15] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Advances in Mathematical Sciences and Applications, 5 (1995), 581-601. [16] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional keller-segel equations, Funkcialaj Ekvacioj-Serio Internacia, 44 (2001), 441-469. [17] K. B. Raper, Dictyostelium discoideum, a new species of slime mold from decaying forest leaves, Journal of Agricultural Research, 50 (1935), 135-147. [18] T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Berlin, 2005. doi: 10.1007/0-8176-4436-9. [19] T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods and Applications of Analysis, 8 (2001), 349-367. [20] T. Senba, T. Nagai and K. Yoshida, Application of the trudinger-moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj-Serio Internacia, 40 (1997), 411-433. [21] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional keller-segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. [22] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic keller-segel system, Journal de Mathématiques Pures et Appliquées, 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020. [23] S. H. Wu, H. Chen and W. X. Li, The local and global existence of the solutions of hyperbolic-parabolic system modeling biological phenomena, Acta Mathematica Scientia, 28 (2008), 101-116. doi: 10.1016/S0252-9602(08)60011-9. [24] S. H. Wu, A free boundary problem for a chemotaxis system, Acta Mathematica Sinica. Chinese Series, 53 (2010), 515-524. [25] S. H. Wu and B. Yue, On existence of local solutions of a moving boundary problem modelling chemotaxis in 1-D, Journal of Partial Differential Equations, 27 (2014), 268-282. [26] Y. Yang, H. Chen, W. A. Liu and B. Sleeman, The solvability of some chemotaxis systems, Journal of Differential Equations, 212 (2005), 432-451. doi: 10.1016/j.jde.2005.01.002.

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##### References:
 [1] H. Chen and S. H. Wu, On existence of solutions for some hyperbolic-parabolic-type chemotaxis systems, IMA Journal of Applied Mathematics, 72 (2007), 331-347. doi: 10.1093/imamat/hxm008. [2] H. Chen and S. H. Wu, The free boundary problem in biological phenomena, Journal of Partial Differential Equations, 20 (2007), 155-168. [3] H. Chen and S. H. Wu, Hyperbolic-parabolic chemotaxis system with nonlinear product terms, Journal of Partial Differential Equations, 21 (2008), 45-58. [4] H. Chen and S. H. Wu, Nonlinear hyperbolic-parabolic system modeling some biological phenomena, Journal of Partial Differential Equations, 24 (2011), 1-14. [5] H. Chen and S. H. Wu, The moving the moving boundary problem in a chemotaxis model, Communications on Pure and Applied Analysis, 11 (2012), 735-746. doi: 10.3934/cpaa.2012.11.735. [6] H. Chen and X. H. Zhong, Norm behaviour of solutions to a parabolic-elliptic system modelling chemotaxis in a domain of $\mathbbR^3$, Mathematical Methods in the Applied Sciences, 27 (2004), 991-1006. doi: 10.1002/mma.479. [7] H. Chen and X. H. Zhong, Global existence and blow-up for the solutions to nonlinear parabolic-elliptic system modelling chemotaxis, IMA Journal of Applied Mathematics, 70 (2005), 221-240. doi: 10.1093/imamat/hxh032. [8] H. Chen and X. H. Zhong, Existence and stability of steady solutions to nonlinear parabolic-elliptic systems modelling chemotaxis, Mathematische Nachrichten, 279 (2006), 1441-1447. doi: 10.1002/mana.200310430. [9] A. Friedman, Free boundary problems in science and technology, Notices of the American Mathematical Society, 47 (2000), 854-861. [10] H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Mathematische Nachrichten, 195 (1998), 77-114. doi: 10.1002/mana.19981950106. [11] T. Hillen and K. J. Painter, A user's guide to pde models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [12] D. Horstmann, From 1970 until present: The keller-segel model in chemotaxis and its consequences I, Jahresbericht der Deutschen Mathematiker Vereinigung, 105 (2003), 103-165. [13] D. Horstmann and G. F. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European Journal of Applied Mathematics, 12 (2001), 159-177. doi: 10.1017/S0956792501004363. [14] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [15] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Advances in Mathematical Sciences and Applications, 5 (1995), 581-601. [16] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional keller-segel equations, Funkcialaj Ekvacioj-Serio Internacia, 44 (2001), 441-469. [17] K. B. Raper, Dictyostelium discoideum, a new species of slime mold from decaying forest leaves, Journal of Agricultural Research, 50 (1935), 135-147. [18] T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Berlin, 2005. doi: 10.1007/0-8176-4436-9. [19] T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods and Applications of Analysis, 8 (2001), 349-367. [20] T. Senba, T. Nagai and K. Yoshida, Application of the trudinger-moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj-Serio Internacia, 40 (1997), 411-433. [21] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional keller-segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. [22] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic keller-segel system, Journal de Mathématiques Pures et Appliquées, 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020. [23] S. H. Wu, H. Chen and W. X. Li, The local and global existence of the solutions of hyperbolic-parabolic system modeling biological phenomena, Acta Mathematica Scientia, 28 (2008), 101-116. doi: 10.1016/S0252-9602(08)60011-9. [24] S. H. Wu, A free boundary problem for a chemotaxis system, Acta Mathematica Sinica. Chinese Series, 53 (2010), 515-524. [25] S. H. Wu and B. Yue, On existence of local solutions of a moving boundary problem modelling chemotaxis in 1-D, Journal of Partial Differential Equations, 27 (2014), 268-282. [26] Y. Yang, H. Chen, W. A. Liu and B. Sleeman, The solvability of some chemotaxis systems, Journal of Differential Equations, 212 (2005), 432-451. doi: 10.1016/j.jde.2005.01.002.
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