Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains

Dingshi Li; Kening Lu; Bixiang Wang; Xiaohu Wang

doi:10.3934/dcds.2018009

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January 2018, 38(1): 187-208. doi: 10.3934/dcds.2018009

Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains

Dingshi Li ^1, , Kening Lu ^2,3, , Bixiang Wang ^4, and Xiaohu Wang ^3,,

1.	School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China
2.	Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA
3.	Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
4.	Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author: Xiaohu Wang, wangxiaohu@scu.edu.cn

Received February 2017 Revised August 2017 Published September 2017

Fund Project: This work was supported by NSFC (11271270, 11601446 and 11201320), NSF (1413603), Excellent Youth Scholars of Sichuan University (2016SCU04A15) and Fundamental Research Funds for the Central Universities (2682015CX059).

In this paper, we study the limiting behavior of dynamics for stochastic reaction-diffusion equations driven by an additive noise and a deterministic non-autonomous forcing on an (n+1)-dimensional thin region when it collapses into an n-dimensional region. We first established the existence of attractors and their properties for these equations on (n+1)-dimensional thin domains. We then show that these attractors converge to the random attractor of the limit equation under the usual semi-distance as the thinness goes to zero.

Keywords: Thin domain, stochastic reaction-diffusion equation, pullback attractor, upper semicontinuity.

Mathematics Subject Classification: Primary:35B40;Secondary:35B41, 37L30.

Citation: Dingshi Li, Kening Lu, Bixiang Wang, Xiaohu Wang. Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 187-208. doi: 10.3934/dcds.2018009

References:

[1]	F. Antoci and M. Prizzi, Reaction-diffusion equations on unbounded thin domains, Topol. Methods Nonlinear Anal., 18 (2001), 283-302. doi: 10.12775/TMNA.2001.035. Google Scholar
[2]	L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998 doi: 10.1007/978-3-662-12878-7. Google Scholar
[3]	J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅰ. Continuity of the set of equilibria, J. Differential Equations, 231 (2006), 551-597. doi: 10.1016/j.jde.2006.06.002. Google Scholar
[4]	J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅱ. The limiting problem, J. Differential Equations, 247 (2009), 174-202. doi: 10.1016/j.jde.2009.03.014. Google Scholar
[5]	J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅲ. Continuity of attractors, J. Differential Equations, 247 (2009), 225-259. doi: 10.1016/j.jde.2008.12.014. Google Scholar
[6]	J. M. Arrieta, A. N. Carvalho, M. C. Pereira and R. P. Da Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal., 74 (2011), 5111-5132. doi: 10.1016/j.na.2011.05.006. Google Scholar
[7]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. Google Scholar
[8]	P. W. Bates, K. 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. Google Scholar
[9]	T. Caraballo, I. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2007), 1489-1507. doi: 10.1137/050647281. Google Scholar
[10]	T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Continuous Dynam. Systems-A, 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415. Google Scholar
[11]	T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Continuous Dynam. Systems-B, 9 (2008), 525-539. doi: 10.3934/dcdsb.2008.9.525. Google Scholar
[12]	I. D. Chueshov and S. Kuksin, Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal., 188 (2008), 117-153. doi: 10.1007/s00205-007-0068-2. Google Scholar
[13]	I. D. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its $α$-approximation, Phys. D, 237 (2008), 1352-1367. doi: 10.1016/j.physd.2008.03.012. Google Scholar
[14]	I. S. Ciuperca, Reaction-diffusion equations on thin domains with varying order of thinness, J. Differential Equations, 126 (1996), 244-291. doi: 10.1006/jdeq.1996.0051. Google Scholar
[15]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar
[16]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar
[17]	T. Elsken, Attractors for reaction-diffusion equations on thin domains whose linear part is non-self-adjoint, J. Differential Equations, 206 (2004), 94-126. doi: 10.1016/j.jde.2004.07.025. Google Scholar
[18]	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. Google Scholar
[19]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. Google Scholar
[20]	J. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95. Google Scholar
[21]	J. Hale and G. Raugel, A reaction-diffusion equation on a thin $L$-shaped domain, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 283-327. doi: 10.1017/S0308210500028043. Google Scholar
[22]	R. Johnson, M. Kamenskii and P. Nistri, Existence of periodic solutions of an autonomous damped wave equation in thin domains, J. Dynam. Differential Equations, 10 (1998), 409-424. doi: 10.1023/A:1022601213052. Google Scholar
[23]	P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. London, Ser. A, 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. Google Scholar
[24]	D. Li, B. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602. doi: 10.1016/j.jde.2016.10.024. Google Scholar
[25]	W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dynam. Differential Equations, 22 (2010), 413-437. doi: 10.1007/s10884-010-9186-x. Google Scholar
[26]	Y. Morita, Stable solutions to the Ginzburg-Landau equation with magnetic effect in a thin domain, Japan J. Indust. Appl. Math., 21 (2004), 129-147. doi: 10.1007/BF03167468. Google Scholar
[27]	M. Prizzi and K. P. Rybakowski, Recent results on thin domain problems, Ⅱ, Topol. Methods Nonlinear Anal., 19 (2002), 199-219. doi: 10.12775/TMNA.2002.010. Google Scholar
[28]	M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 237 (2001), 271-320. doi: 10.1006/jdeq.2000.3917. Google Scholar
[29]	G. Raugel and G. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568. doi: 10.2307/2152776. Google Scholar
[30]	D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces, Comm. Math. Phys., 93 (1984), 285-300. doi: 10.1007/BF01258529. Google Scholar
[31]	B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185-192. Google Scholar
[32]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997 doi: 10.1007/978-1-4612-0645-3. Google Scholar
[33]	S. M. Ulam and J. von Neumann, Random ergodic theorems, Bull. Amer. Math. Soc. , 51 (1945), p660. Google Scholar
[34]	B. Wang, Suffcient 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. Google Scholar
[35]	B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Continuous Dynam. Systems-A, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar
[36]	X. Wang, K. Lu and B. Wang, Long term behavior of delay parabolic equations with additive noise and deterministic time dependent forcing, SIAM J. Appl. Dynam. Syst., 14 (2015), 1018-1047. doi: 10.1137/140991819. Google Scholar

show all references

References:

[1]	F. Antoci and M. Prizzi, Reaction-diffusion equations on unbounded thin domains, Topol. Methods Nonlinear Anal., 18 (2001), 283-302. doi: 10.12775/TMNA.2001.035. Google Scholar
[2]	L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998 doi: 10.1007/978-3-662-12878-7. Google Scholar
[3]	J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅰ. Continuity of the set of equilibria, J. Differential Equations, 231 (2006), 551-597. doi: 10.1016/j.jde.2006.06.002. Google Scholar
[4]	J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅱ. The limiting problem, J. Differential Equations, 247 (2009), 174-202. doi: 10.1016/j.jde.2009.03.014. Google Scholar
[5]	J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅲ. Continuity of attractors, J. Differential Equations, 247 (2009), 225-259. doi: 10.1016/j.jde.2008.12.014. Google Scholar
[6]	J. M. Arrieta, A. N. Carvalho, M. C. Pereira and R. P. Da Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal., 74 (2011), 5111-5132. doi: 10.1016/j.na.2011.05.006. Google Scholar
[7]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. Google Scholar
[8]	P. W. Bates, K. 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. Google Scholar
[9]	T. Caraballo, I. D. Chueshov and P. E. Kloeden, Synchronization of a stochastic reaction-diffusion system on a thin two-layer domain, SIAM J. Math. Anal., 38 (2007), 1489-1507. doi: 10.1137/050647281. Google Scholar
[10]	T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Continuous Dynam. Systems-A, 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415. Google Scholar
[11]	T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Continuous Dynam. Systems-B, 9 (2008), 525-539. doi: 10.3934/dcdsb.2008.9.525. Google Scholar
[12]	I. D. Chueshov and S. Kuksin, Random kick-forced 3D Navier-Stokes equations in a thin domain, Arch. Ration. Mech. Anal., 188 (2008), 117-153. doi: 10.1007/s00205-007-0068-2. Google Scholar
[13]	I. D. Chueshov and S. Kuksin, Stochastic 3D Navier-Stokes equations in a thin domain and its $α$-approximation, Phys. D, 237 (2008), 1352-1367. doi: 10.1016/j.physd.2008.03.012. Google Scholar
[14]	I. S. Ciuperca, Reaction-diffusion equations on thin domains with varying order of thinness, J. Differential Equations, 126 (1996), 244-291. doi: 10.1006/jdeq.1996.0051. Google Scholar
[15]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar
[16]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar
[17]	T. Elsken, Attractors for reaction-diffusion equations on thin domains whose linear part is non-self-adjoint, J. Differential Equations, 206 (2004), 94-126. doi: 10.1016/j.jde.2004.07.025. Google Scholar
[18]	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. Google Scholar
[19]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. Google Scholar
[20]	J. Hale and G. Raugel, Reaction-diffusion equations on thin domains, J. Math. Pures Appl., 71 (1992), 33-95. Google Scholar
[21]	J. Hale and G. Raugel, A reaction-diffusion equation on a thin $L$-shaped domain, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 283-327. doi: 10.1017/S0308210500028043. Google Scholar
[22]	R. Johnson, M. Kamenskii and P. Nistri, Existence of periodic solutions of an autonomous damped wave equation in thin domains, J. Dynam. Differential Equations, 10 (1998), 409-424. doi: 10.1023/A:1022601213052. Google Scholar
[23]	P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. London, Ser. A, 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. Google Scholar
[24]	D. Li, B. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602. doi: 10.1016/j.jde.2016.10.024. Google Scholar
[25]	W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dynam. Differential Equations, 22 (2010), 413-437. doi: 10.1007/s10884-010-9186-x. Google Scholar
[26]	Y. Morita, Stable solutions to the Ginzburg-Landau equation with magnetic effect in a thin domain, Japan J. Indust. Appl. Math., 21 (2004), 129-147. doi: 10.1007/BF03167468. Google Scholar
[27]	M. Prizzi and K. P. Rybakowski, Recent results on thin domain problems, Ⅱ, Topol. Methods Nonlinear Anal., 19 (2002), 199-219. doi: 10.12775/TMNA.2002.010. Google Scholar
[28]	M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations, 237 (2001), 271-320. doi: 10.1006/jdeq.2000.3917. Google Scholar
[29]	G. Raugel and G. Sell, Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc., 6 (1993), 503-568. doi: 10.2307/2152776. Google Scholar
[30]	D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces, Comm. Math. Phys., 93 (1984), 285-300. doi: 10.1007/BF01258529. Google Scholar
[31]	B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, (1992), 185-192. Google Scholar
[32]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997 doi: 10.1007/978-1-4612-0645-3. Google Scholar
[33]	S. M. Ulam and J. von Neumann, Random ergodic theorems, Bull. Amer. Math. Soc. , 51 (1945), p660. Google Scholar
[34]	B. Wang, Suffcient 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. Google Scholar
[35]	B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Continuous Dynam. Systems-A, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar
[36]	X. Wang, K. Lu and B. Wang, Long term behavior of delay parabolic equations with additive noise and deterministic time dependent forcing, SIAM J. Appl. Dynam. Syst., 14 (2015), 1018-1047. doi: 10.1137/140991819. Google Scholar

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Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains

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