October  2018, 38(10): 5067-5083. doi: 10.3934/dcds.2018222

Chemotaxis model with nonlocal nonlinear reaction in the whole space

1. 

Beijing University of Chemical Technology, 100029, Beijing, China

2. 

Universität Mannheim, 68131, Mannheim, Germany

* Corresponding author: Shen Bian

Received  November 2017 Revised  February 2018 Published  July 2018

Fund Project: The first author is supported by NNSF of China (No. 11501025) and the Alexander von Humboldt Foundation, the second author is supported by DFG Project CH 955/3-1.

This paper deals with a parabolic-elliptic chemotaxis system with nonlocal type of source in the whole space. It's proved that the initial value problem possesses a unique global solution which is uniformly bounded. Here we identify the exponents regimes of nonlinear reaction and aggregation in such a way that their scaling and the diffusion term coincide (see Introduction). Comparing to the classical KS model (without the source term), it's shown that how energy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solution without any restriction on the initial data.

Citation: Shen Bian, Li Chen, Evangelos A. Latos. Chemotaxis model with nonlocal nonlinear reaction in the whole space. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5067-5083. doi: 10.3934/dcds.2018222
References:
[1]

S. Bian and L. Chen, A nonlocal reaction diffusion equation and its relation with Fujita exponent, Journal of Mathematical Analysis and Applications, 444 (2016), 1479-1489.  doi: 10.1016/j.jmaa.2016.07.014.

[2]

S. Bian and J. G. Liu, Dynamic and steady states for multi-dimensional keller-segel model with diffusion exponent m > 0, Comm. Math. Phys, 323 (2013), 1017-1070.  doi: 10.1007/s00220-013-1777-z.

[3]

A. BlanchetJ. A. Carrillo and P. Laurencot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var., 35 (2009), 133-168.  doi: 10.1007/s00526-008-0200-7.

[4]

A. BlanchetJ. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Eletron. J. Differ. Equ., 44 (2006), 1-33. 

[5]

R. Fisher, The wave of advance of advantageous genes, Ann Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[6]

H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = ∆u +u1+α, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 13 (1966), 109-124. 

[7]

E. GalakhovO. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.

[8]

K. Hayaka, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505.  doi: 10.3792/pja/1195519254.

[9]

X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.  doi: 10.1016/j.jmaa.2015.12.058.

[10]

B. Hu and Y. Tao, Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7.  doi: 10.1016/j.aml.2016.08.003.

[11]

T. B. Issa and W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), 926-973.  doi: 10.1137/16M1092428.

[12]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 239 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[13]

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

[14]

A. Kolmogorov, I. Petrovsky and N. Piskunov, Investigation of the equation of diffusion combined with increasing of the substance and its application to a biology problem, Bull Moscow State Univ Ser A: Math and Mech., 1 (1937), 1-25.

[15]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl. 23, Providence, RI, 1968.

[16]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics. V. 14, American Mathematical Society Providence, Rhode Island, 1997. doi: 10.1090/gsm/014.

[17]

M. Negreanu and J. Tello, On a competitive system under chemotactic effects with non-local terms, Nonlinearity, 26 (2013), 1083-1103.  doi: 10.1088/0951-7715/26/4/1083.

[18]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[19]

B. Perthame, Transport Equations in Biology, Birkhaeuser Verlag, Basel-Boston-Berlin, 2007.

[20]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate keller-segel model with a power factor in drift term, J. Diff. Eqns., 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.

[21]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[22]

Y. Wang and J. Liu, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Nonlinear Analysis: Real World Applications, 38 (2017), 113-130.  doi: 10.1016/j.nonrwa.2017.04.010.

[23]

L. WangC. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Diff. Eqns., 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.

[24]

Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z.

[25]

J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140.  doi: 10.1016/j.jde.2015.02.003.

show all references

References:
[1]

S. Bian and L. Chen, A nonlocal reaction diffusion equation and its relation with Fujita exponent, Journal of Mathematical Analysis and Applications, 444 (2016), 1479-1489.  doi: 10.1016/j.jmaa.2016.07.014.

[2]

S. Bian and J. G. Liu, Dynamic and steady states for multi-dimensional keller-segel model with diffusion exponent m > 0, Comm. Math. Phys, 323 (2013), 1017-1070.  doi: 10.1007/s00220-013-1777-z.

[3]

A. BlanchetJ. A. Carrillo and P. Laurencot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var., 35 (2009), 133-168.  doi: 10.1007/s00526-008-0200-7.

[4]

A. BlanchetJ. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Eletron. J. Differ. Equ., 44 (2006), 1-33. 

[5]

R. Fisher, The wave of advance of advantageous genes, Ann Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[6]

H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = ∆u +u1+α, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 13 (1966), 109-124. 

[7]

E. GalakhovO. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.

[8]

K. Hayaka, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505.  doi: 10.3792/pja/1195519254.

[9]

X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.  doi: 10.1016/j.jmaa.2015.12.058.

[10]

B. Hu and Y. Tao, Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7.  doi: 10.1016/j.aml.2016.08.003.

[11]

T. B. Issa and W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), 926-973.  doi: 10.1137/16M1092428.

[12]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 239 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[13]

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

[14]

A. Kolmogorov, I. Petrovsky and N. Piskunov, Investigation of the equation of diffusion combined with increasing of the substance and its application to a biology problem, Bull Moscow State Univ Ser A: Math and Mech., 1 (1937), 1-25.

[15]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl. 23, Providence, RI, 1968.

[16]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics. V. 14, American Mathematical Society Providence, Rhode Island, 1997. doi: 10.1090/gsm/014.

[17]

M. Negreanu and J. Tello, On a competitive system under chemotactic effects with non-local terms, Nonlinearity, 26 (2013), 1083-1103.  doi: 10.1088/0951-7715/26/4/1083.

[18]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[19]

B. Perthame, Transport Equations in Biology, Birkhaeuser Verlag, Basel-Boston-Berlin, 2007.

[20]

Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate keller-segel model with a power factor in drift term, J. Diff. Eqns., 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.

[21]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[22]

Y. Wang and J. Liu, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Nonlinear Analysis: Real World Applications, 38 (2017), 113-130.  doi: 10.1016/j.nonrwa.2017.04.010.

[23]

L. WangC. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Diff. Eqns., 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.

[24]

Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z.

[25]

J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140.  doi: 10.1016/j.jde.2015.02.003.

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