November  2018, 23(9): 4003-4020. doi: 10.3934/dcdsb.2018121

Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

Received  June 2017 Revised  November 2017 Published  November 2018 Early access  April 2018

Fund Project: The work is partially supported by PRC grant NSFC 11771380, 11401515.

In this paper, we study the generalized chemotaxis system with fractional Laplacian. The existence and the uniqueness of global classical solution are proved under the assumption that the initial data are small enough. During the proof, with the help of the fixed point theorem, the asymptotic decay behaviors of $ u $ and $ \nabla{v} $ are also shown.

Citation: Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121
References:
[1]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[2]

P. BilerT. Funaki and W. A. Woyczy$ \acute{n} $ski, Interacting particle approximation for nonlocal quadratic evolution problems, Probab. Math. Statist., 19 (1999), 267-286. 

[3]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. 

[4]

P. Biler and W. A. Woyczy$ \acute{n} $ski, Global and exploding solutions for nonlocal quardratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869. 

[5]

P. Biler and W. A. Woyczy$ \acute{n} $ski, Nonlocal quadratic evolution problems, Banach Center Publ., 52 (2000), 11-24. 

[6]

P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262.  doi: 10.1007/s00028-009-0048-0.

[7]

P. Biler, T Cie$ \acute{s} $lak, G. Karch and J. Zienkiewicz, Local criteria for blowup in two-dimensional chemotaxis models, Discrete & Continuous Dynamical Systems - A, 37 (2017), 1841-1856, arXiv: 1410.7870. doi: 10.3934/dcds.2017077.

[8]

H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304.  doi: 10.1007/BF02790212.

[9]

L. Corrias and B. Perthame, Critical space for the parabolic-parabolic Keller-Segel model in $ \mathbb{R}^{n} $, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 745-750.  doi: 10.1016/j.crma.2006.03.008.

[10]

L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces, Math. Comput. Modelling, 47 (2008), 755-764.  doi: 10.1016/j.mcm.2007.06.005.

[11]

C. Escudero, The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918.  doi: 10.1088/0951-7715/19/12/010.

[12]

B. L. Guo, X. K. Pu and F. H. Huang, Fractional Partial differential Equations and their Numerical Solutions, Originally published by Science Press in 2011, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.

[13]

E. F. Keller and L. A. Segel, Initiation of smile mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. 

[14]

D. Li and J.L. Rodrigo, Finite-time singularities of an aggregation equation in $ \mathbb{R}^{n} $ with fractional dissipation, Comm. Math. Phys., 287 (2009), 687-703.  doi: 10.1007/s00220-008-0669-0.

[15]

D. Li and J. L. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation, Adv. in Math., 220 (2009), 1717-1738.  doi: 10.1016/j.aim.2008.10.016.

[16]

D. LiJ. L. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, Rev. Mat. Iberoamericana, 26 (2010), 295-332. 

[17]

D. Q. Li and Y. M. Chen, Nonlinear Evolution Equation, Science Press, 1999.

[18]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

show all references

References:
[1]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[2]

P. BilerT. Funaki and W. A. Woyczy$ \acute{n} $ski, Interacting particle approximation for nonlocal quadratic evolution problems, Probab. Math. Statist., 19 (1999), 267-286. 

[3]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. 

[4]

P. Biler and W. A. Woyczy$ \acute{n} $ski, Global and exploding solutions for nonlocal quardratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869. 

[5]

P. Biler and W. A. Woyczy$ \acute{n} $ski, Nonlocal quadratic evolution problems, Banach Center Publ., 52 (2000), 11-24. 

[6]

P. Biler and G. Karch, Blowup of solutions to generalized Keller-Segel model, J. Evol. Equ., 10 (2010), 247-262.  doi: 10.1007/s00028-009-0048-0.

[7]

P. Biler, T Cie$ \acute{s} $lak, G. Karch and J. Zienkiewicz, Local criteria for blowup in two-dimensional chemotaxis models, Discrete & Continuous Dynamical Systems - A, 37 (2017), 1841-1856, arXiv: 1410.7870. doi: 10.3934/dcds.2017077.

[8]

H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304.  doi: 10.1007/BF02790212.

[9]

L. Corrias and B. Perthame, Critical space for the parabolic-parabolic Keller-Segel model in $ \mathbb{R}^{n} $, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 745-750.  doi: 10.1016/j.crma.2006.03.008.

[10]

L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces, Math. Comput. Modelling, 47 (2008), 755-764.  doi: 10.1016/j.mcm.2007.06.005.

[11]

C. Escudero, The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918.  doi: 10.1088/0951-7715/19/12/010.

[12]

B. L. Guo, X. K. Pu and F. H. Huang, Fractional Partial differential Equations and their Numerical Solutions, Originally published by Science Press in 2011, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.

[13]

E. F. Keller and L. A. Segel, Initiation of smile mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. 

[14]

D. Li and J.L. Rodrigo, Finite-time singularities of an aggregation equation in $ \mathbb{R}^{n} $ with fractional dissipation, Comm. Math. Phys., 287 (2009), 687-703.  doi: 10.1007/s00220-008-0669-0.

[15]

D. Li and J. L. Rodrigo, Refined blowup criteria and nonsymmetric blowup of an aggregation equation, Adv. in Math., 220 (2009), 1717-1738.  doi: 10.1016/j.aim.2008.10.016.

[16]

D. LiJ. L. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, Rev. Mat. Iberoamericana, 26 (2010), 295-332. 

[17]

D. Q. Li and Y. M. Chen, Nonlinear Evolution Equation, Science Press, 1999.

[18]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

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