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November  2018, 17(6): 2577-2592. doi: 10.3934/cpaa.2018122

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

 1 School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China 2 School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, China

* Corresponding author

Received  February 2018 Revised  March 2018 Published  June 2018

Fund Project: This work is supported by National Natural Science Foundation of China (Grant No. 11131005) and the Fundamental Research Funds for the Central Universities (Grant No. 2014201020202).

In this paper, we study a free boundary problem for a class of parabolic-elliptic type chemotaxis model in high dimensional symmetry domain Ω. 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 Ω with free boundary condition. Besides, we get the explicit formula for the free boundary and show the chemotactic collapse for the solution of the system.

Citation: Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122
References:
 [1] H. Chen, W. B. Lv and S. H. Wu, A free boundary problem for a class of parabolic type chemotaxis model, Kinetic and Related Models, 8 (2015), 667-684.   Google Scholar [2] H. Chen, W. B. Lv and S. H. Wu, Solvability of a parabolic-hyperbolic type chemotaxis system in 1-dimensional domain, Acta Mathematics Scientia, Series B, English Edition, 36 (2016), 1285-1304.   Google Scholar [3] H. Chen and S. H. Wu, The free boundary problem in biological phenomena, Journal of Partial Differential Equations, 20 (2007), 155-168.   Google Scholar [4] 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.   Google Scholar [5] H. Chen and S. H. Wu, Hyperbolic-parabolic chemotaxis system with nonlinear product terms, Journal of Partial Differential Equations, 21 (2008), 45-58.   Google Scholar [6] H. Chen and S. H. Wu, Nonlinear hyperbolic-parabolic system modeling some biological phenomena, Journal of Partial Differential Equations, 24 (2011), 1-14.   Google Scholar [7] H. Chen and S. H. Wu, The moving boundary problem in a chemotaxis model, Communications on Pure and Applied Analysis, 11 (2012), 735-746.   Google Scholar [8] H. Chen and X. H. Zhong, Norm behaviour of solutions to a parabolic-elliptic system modelling chemotaxis in a domain of $\mathbb{R}^3$, Mathematical Methods in the Applied Sciences, 27 (2004), 991-1006.   Google Scholar [9] 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.   Google Scholar [10] 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.   Google Scholar [11] T. Cieślak and P. Laurençot, Finite time blow-up for radially symmetric solutions to a critical quasilinear smoluchowski-poisson system, Comptes Rendus Mathematique, 347 (2009), 237-242.   Google Scholar [12] A. Friedman, Free boundary problems in science and technology, Notices of the American Mathematical Society, 47 (2000), 854-861.   Google Scholar [13] M. A. Herrero, Asymptotic properties of reaction-diffusion systems modeling chemotaxis, In Applied and Industrial Mathematics, Venice2, 1998, pages 89-108. Springer, 2000.  Google Scholar [14] M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.   Google Scholar [15] M. A. Herrero, E. Medina and J. J. L. Velázquez, Self-similar blow-up for a reaction-diffusion system, Journal of Computational and Applied Mathematics, 97 (1998), 99-119.   Google Scholar [16] T. Hillen and K. J. Painter, A user's guide to pde models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.   Google Scholar [17] D. Horstmann, From 1970 until present: the keller-segel model in chemotaxis and its consequences 1, Jahresberichte DMV, 105 (2003), 103-165.   Google Scholar [18] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Transactions of the American Mathematical Society, 329 (1992), 819-824.   Google Scholar [19] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415.   Google Scholar [20] K. B. Raper, Dictyostelium discoideum, a new species of slime mold from decaying forest leaves, Journal of Agricultural Research, 50 (1935), 135-147.   Google Scholar [21] M. Taylor, Partial Differential Equations Ⅲ, volume 116. Springer Science and Business Media, 2013.  Google Scholar [22] S. H. Wu, A free boundary problem for a chemotaxis system, Acta Mathematica Sinica. Chinese Series, 53 (2010), 515-524.   Google Scholar [23] 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.   Google Scholar [24] 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, Series B, English Edition, 28 (2008), 101-116.   Google Scholar [25] Y. Yang, H. Chen, W. A. Liu and B. Sleeman, The solvability of some chemotaxis systems, Journal of Differential Equations, 212 (2005), 432-451.   Google Scholar

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References:
 [1] H. Chen, W. B. Lv and S. H. Wu, A free boundary problem for a class of parabolic type chemotaxis model, Kinetic and Related Models, 8 (2015), 667-684.   Google Scholar [2] H. Chen, W. B. Lv and S. H. Wu, Solvability of a parabolic-hyperbolic type chemotaxis system in 1-dimensional domain, Acta Mathematics Scientia, Series B, English Edition, 36 (2016), 1285-1304.   Google Scholar [3] H. Chen and S. H. Wu, The free boundary problem in biological phenomena, Journal of Partial Differential Equations, 20 (2007), 155-168.   Google Scholar [4] 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.   Google Scholar [5] H. Chen and S. H. Wu, Hyperbolic-parabolic chemotaxis system with nonlinear product terms, Journal of Partial Differential Equations, 21 (2008), 45-58.   Google Scholar [6] H. Chen and S. H. Wu, Nonlinear hyperbolic-parabolic system modeling some biological phenomena, Journal of Partial Differential Equations, 24 (2011), 1-14.   Google Scholar [7] H. Chen and S. H. Wu, The moving boundary problem in a chemotaxis model, Communications on Pure and Applied Analysis, 11 (2012), 735-746.   Google Scholar [8] H. Chen and X. H. Zhong, Norm behaviour of solutions to a parabolic-elliptic system modelling chemotaxis in a domain of $\mathbb{R}^3$, Mathematical Methods in the Applied Sciences, 27 (2004), 991-1006.   Google Scholar [9] 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.   Google Scholar [10] 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.   Google Scholar [11] T. Cieślak and P. Laurençot, Finite time blow-up for radially symmetric solutions to a critical quasilinear smoluchowski-poisson system, Comptes Rendus Mathematique, 347 (2009), 237-242.   Google Scholar [12] A. Friedman, Free boundary problems in science and technology, Notices of the American Mathematical Society, 47 (2000), 854-861.   Google Scholar [13] M. A. Herrero, Asymptotic properties of reaction-diffusion systems modeling chemotaxis, In Applied and Industrial Mathematics, Venice2, 1998, pages 89-108. Springer, 2000.  Google Scholar [14] M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.   Google Scholar [15] M. A. Herrero, E. Medina and J. J. L. Velázquez, Self-similar blow-up for a reaction-diffusion system, Journal of Computational and Applied Mathematics, 97 (1998), 99-119.   Google Scholar [16] T. Hillen and K. J. Painter, A user's guide to pde models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217.   Google Scholar [17] D. Horstmann, From 1970 until present: the keller-segel model in chemotaxis and its consequences 1, Jahresberichte DMV, 105 (2003), 103-165.   Google Scholar [18] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Transactions of the American Mathematical Society, 329 (1992), 819-824.   Google Scholar [19] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415.   Google Scholar [20] K. B. Raper, Dictyostelium discoideum, a new species of slime mold from decaying forest leaves, Journal of Agricultural Research, 50 (1935), 135-147.   Google Scholar [21] M. Taylor, Partial Differential Equations Ⅲ, volume 116. Springer Science and Business Media, 2013.  Google Scholar [22] S. H. Wu, A free boundary problem for a chemotaxis system, Acta Mathematica Sinica. Chinese Series, 53 (2010), 515-524.   Google Scholar [23] 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.   Google Scholar [24] 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, Series B, English Edition, 28 (2008), 101-116.   Google Scholar [25] Y. Yang, H. Chen, W. A. Liu and B. Sleeman, The solvability of some chemotaxis systems, Journal of Differential Equations, 212 (2005), 432-451.   Google Scholar
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