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March  2019, 24(3): 1367-1391. doi: 10.3934/dcdsb.2019020

Asymptotic behavior of the stochastic Keller-Segel equations

1. 

School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China

2. 

Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88001, USA

3. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author

Received  December 2017 Revised  March 2018 Published  March 2019 Early access  January 2019

Fund Project: The second author was supported by NSF (DMS-1446139), NIH (U54CA132383) and NNSF of China (No.11371048).

This paper deals with the asymptotic behavior of the solutions of the non-autonomous one-dimensional stochastic Keller-Segel equations defined in a bounded interval with Neumann boundary conditions. We prove the existence and uniqueness of tempered pullback random attractors under certain conditions. We also establish the convergence of the solutions as well as the pullback random attractors of the stochastic equations as the intensity of noise approaches zero.

Citation: Yadong Shang, Jianjun Paul Tian, Bixiang Wang. Asymptotic behavior of the stochastic Keller-Segel equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1367-1391. doi: 10.3934/dcdsb.2019020
References:
[1]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.  doi: 10.3934/dcdsb.2013.18.643.

[2]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[4]

W. J. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.

[5]

T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.

[6]

T. CaraballoJ. Real and I. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.

[7]

T. Caraballo and J. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513. 

[8]

T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.

[9]

T. CaraballoM. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.

[10]

T. CaraballoJ. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.

[11]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.  doi: 10.1080/1468936042000207792.

[12]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[13]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[14]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.  doi: 10.4310/CMS.2003.v1.n1.a9.

[15]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[16]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modeling chemotaxis, Math. Nachr., 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.

[17]

M. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681.  doi: 10.1007/s10884-011-9222-5.

[18]

M. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.  doi: 10.1142/S0219493711003358.

[19]

M. Garrido-AtienzaB. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, International J. Bifurcation and Chaos, 20 (2010), 2761-2782.  doi: 10.1142/S0218127410027349.

[20]

B. GessW. Liu and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.

[21]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dyn. Diff. Eqns., 25 (2013), 121-157.  doi: 10.1007/s10884-013-9294-5.

[22]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.

[23]

M. A. Herrero and J. J. L. Velazquez, A blow-up mechanism for a chemotaxis model, Ann. Scoula. Norm. Sup. Pisa IV, 24 (1997), 633-683. 

[24]

D. Horstmann and G. Wang, Blowup in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[25]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882.  doi: 10.3934/dcds.2009.24.855.

[26]

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

[27]

P. E. Kloeden and J. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A., 463 (2007), 163–181. doi: 10.1098/rspa.2006.1753.

[28]

I. R. Lapidus and M. Levandowsky, Modeling chemosensory responses of swimming eukaryotes, biological growth and spread, Lecture Notes in Biomathematics, 38 (1980), 388-396. 

[29]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.

[30]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[31]

T. NagaiT. Senba and T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J., 30 (2000), 463-497.  doi: 10.32917/hmj/1206124609.

[32]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[33]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, TMA, 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.

[34]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.  doi: 10.1090/S0002-9947-1985-0808736-1.

[35]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185–192, Dresden, 1992.

[36]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[37]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $ \mathbb{R} ^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[38]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[39]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31pp. doi: 10.1142/S0219493714500099.

[40]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[41]

A. Yagi, Norm behavior of solutions to the parabolic system of chemotaxis, Math. Japonica, 45 (1997), 241-265. 

show all references

References:
[1]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666.  doi: 10.3934/dcdsb.2013.18.643.

[2]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[4]

W. J. BeynB. GessP. Lescot and M. Röckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469.  doi: 10.1080/03605302.2010.523919.

[5]

T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.

[6]

T. CaraballoJ. Real and I. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.

[7]

T. Caraballo and J. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513. 

[8]

T. CaraballoM. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.

[9]

T. CaraballoM. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.

[10]

T. CaraballoJ. LangaV. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems, Set-Valued Analysis, 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.

[11]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144.  doi: 10.1080/1468936042000207792.

[12]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[13]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[14]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.  doi: 10.4310/CMS.2003.v1.n1.a9.

[15]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[16]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modeling chemotaxis, Math. Nachr., 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.

[17]

M. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681.  doi: 10.1007/s10884-011-9222-5.

[18]

M. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.  doi: 10.1142/S0219493711003358.

[19]

M. Garrido-AtienzaB. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, International J. Bifurcation and Chaos, 20 (2010), 2761-2782.  doi: 10.1142/S0218127410027349.

[20]

B. GessW. Liu and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253.  doi: 10.1016/j.jde.2011.02.013.

[21]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dyn. Diff. Eqns., 25 (2013), 121-157.  doi: 10.1007/s10884-013-9294-5.

[22]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559.  doi: 10.1016/j.jde.2013.04.023.

[23]

M. A. Herrero and J. J. L. Velazquez, A blow-up mechanism for a chemotaxis model, Ann. Scoula. Norm. Sup. Pisa IV, 24 (1997), 633-683. 

[24]

D. Horstmann and G. Wang, Blowup in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.

[25]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst., 24 (2009), 855-882.  doi: 10.3934/dcds.2009.24.855.

[26]

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

[27]

P. E. Kloeden and J. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Serie A., 463 (2007), 163–181. doi: 10.1098/rspa.2006.1753.

[28]

I. R. Lapidus and M. Levandowsky, Modeling chemosensory responses of swimming eukaryotes, biological growth and spread, Lecture Notes in Biomathematics, 38 (1980), 388-396. 

[29]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.

[30]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[31]

T. NagaiT. Senba and T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J., 30 (2000), 463-497.  doi: 10.32917/hmj/1206124609.

[32]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[33]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, TMA, 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.

[34]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.  doi: 10.1090/S0002-9947-1985-0808736-1.

[35]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185–192, Dresden, 1992.

[36]

B. Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[37]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $ \mathbb{R} ^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[38]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[39]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31pp. doi: 10.1142/S0219493714500099.

[40]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[41]

A. Yagi, Norm behavior of solutions to the parabolic system of chemotaxis, Math. Japonica, 45 (1997), 241-265. 

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