Open Access
2013 Random Attractors for Stochastic Retarded Reaction-Diffusion Equations on Unbounded Domains
Xiaoquan Ding, Jifa Jiang
Abstr. Appl. Anal. 2013: 1-16 (2013). DOI: 10.1155/2013/981576
Abstract

This paper is devoted to a stochastic retarded reaction-diffusion equation on all d -dimensional space with additive white noise. We first show that the stochastic retarded reaction-diffusion equation generates a random dynamical system by transforming this stochastic equation into a random one through a tempered stationary random homeomorphism. Then, we establish the existence of a random attractor for the random equation. And the existence of a random attractor for the stochastic equation follows from the conjugation relation between two random dynamical systems. The pullback asymptotic compactness is proved by uniform estimates on solutions for large space and time variables. These estimates are obtained by a cut-off technique.

References

1.

H. Crauel and F. Flandoli, “Attractors for random dynamical systems,” Probability Theory and Related Fields, vol. 100, no. 3, pp. 365–393, 1994. MR1305587 0819.58023 10.1007/BF01193705 H. Crauel and F. Flandoli, “Attractors for random dynamical systems,” Probability Theory and Related Fields, vol. 100, no. 3, pp. 365–393, 1994. MR1305587 0819.58023 10.1007/BF01193705

2.

F. Flandoli and B. Schmalfuss, “Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise,” Stochastics and Stochastics Reports, vol. 59, no. 1-2, pp. 21–45, 1996. MR1427258 10.1080/17442509608834083 F. Flandoli and B. Schmalfuss, “Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise,” Stochastics and Stochastics Reports, vol. 59, no. 1-2, pp. 21–45, 1996. MR1427258 10.1080/17442509608834083

3.

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, USA, 1988. MR941371 J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, USA, 1988. MR941371

4.

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, The Netherlands, 1992. MR1156492 0850.68019 A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, The Netherlands, 1992. MR1156492 0850.68019

5.

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, NY, USA, 2nd edition, 1997. MR1441312 R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, NY, USA, 2nd edition, 1997. MR1441312

6.

H. Crauel, A. Debussche, and F. Flandoli, “Random attractors,” Journal of Dynamics and Differential Equations, vol. 9, no. 2, pp. 307–341, 1997. MR1451294 0884.58064 10.1007/BF02219225 H. Crauel, A. Debussche, and F. Flandoli, “Random attractors,” Journal of Dynamics and Differential Equations, vol. 9, no. 2, pp. 307–341, 1997. MR1451294 0884.58064 10.1007/BF02219225

7.

X. Fan, “Random attractors for damped stochastic wave equations with multiplicative noise,” International Journal of Mathematics, vol. 19, no. 4, pp. 421–437, 2008. MR2416723 1148.37038 10.1142/S0129167X08004741 X. Fan, “Random attractors for damped stochastic wave equations with multiplicative noise,” International Journal of Mathematics, vol. 19, no. 4, pp. 421–437, 2008. MR2416723 1148.37038 10.1142/S0129167X08004741

8.

Y. Li and B. Guo, “Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations,” Journal of Differential Equations, vol. 245, no. 7, pp. 1775–1800, 2008. MR2433486 1188.37076 10.1016/j.jde.2008.06.031 Y. Li and B. Guo, “Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations,” Journal of Differential Equations, vol. 245, no. 7, pp. 1775–1800, 2008. MR2433486 1188.37076 10.1016/j.jde.2008.06.031

9.

P. W. Bates, K. Lu, and B. Wang, “Random attractors for stochastic reaction-diffusion equations on unbounded domains,” Journal of Differential Equations, vol. 246, no. 2, pp. 845–869, 2009. MR2468738 1155.35112 10.1016/j.jde.2008.05.017 P. W. Bates, K. Lu, and B. Wang, “Random attractors for stochastic reaction-diffusion equations on unbounded domains,” Journal of Differential Equations, vol. 246, no. 2, pp. 845–869, 2009. MR2468738 1155.35112 10.1016/j.jde.2008.05.017

10.

B. Wang, “Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains,” Journal of Differential Equations, vol. 246, no. 6, pp. 2506–2537, 2009. MR2498851 1165.60025 10.1016/j.jde.2008.10.012 B. Wang, “Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains,” Journal of Differential Equations, vol. 246, no. 6, pp. 2506–2537, 2009. MR2498851 1165.60025 10.1016/j.jde.2008.10.012

11.

B. Wang, “Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2811–2828, 2009. MR2532807 1173.37065 B. Wang, “Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2811–2828, 2009. MR2532807 1173.37065

12.

Z. Shen, S. Zhou, and W. Shen, “One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation,” Journal of Differential Equations, vol. 248, no. 6, pp. 1432–1457, 2010. MR2593049 1190.60047 10.1016/j.jde.2009.10.007 Z. Shen, S. Zhou, and W. Shen, “One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation,” Journal of Differential Equations, vol. 248, no. 6, pp. 1432–1457, 2010. MR2593049 1190.60047 10.1016/j.jde.2009.10.007

13.

B. Wang, “Asymptotic behavior of stochastic wave equations with critical exponents on ${\mathbb{R}}^{3}$,” Transactions of the American Mathematical Society, vol. 363, no. 7, pp. 3639–3663, 2011. MR2775822 1230.37095 10.1090/S0002-9947-2011-05247-5 B. Wang, “Asymptotic behavior of stochastic wave equations with critical exponents on ${\mathbb{R}}^{3}$,” Transactions of the American Mathematical Society, vol. 363, no. 7, pp. 3639–3663, 2011. MR2775822 1230.37095 10.1090/S0002-9947-2011-05247-5

14.

B. Gess, “Random attractors for degenerate stochastic partial differential equations,” Journal of Dynamics and Differential Equations, vol. 25, no. 1, pp. 121–157, 2013. MR3027636 10.1007/s10884-013-9294-5 B. Gess, “Random attractors for degenerate stochastic partial differential equations,” Journal of Dynamics and Differential Equations, vol. 25, no. 1, pp. 121–157, 2013. MR3027636 10.1007/s10884-013-9294-5

15.

B. Gess, W. Liu, and M. Röckner, “Random attractors for a class of stochastic partial differential equations driven by general additive noise,” Journal of Differential Equations, vol. 251, no. 4-5, pp. 1225–1253, 2011. MR2812588 1228.35062 10.1016/j.jde.2011.02.013 B. Gess, W. Liu, and M. Röckner, “Random attractors for a class of stochastic partial differential equations driven by general additive noise,” Journal of Differential Equations, vol. 251, no. 4-5, pp. 1225–1253, 2011. MR2812588 1228.35062 10.1016/j.jde.2011.02.013

16.

W.-J. Beyn, B. Gess, P. Lescot, and M. Röckner, “The global random attractor for a class of stochastic porous media equations,” Communications in Partial Differential Equations, vol. 36, no. 3, pp. 446–469, 2011. MR2763333 10.1080/03605302.2010.523919 W.-J. Beyn, B. Gess, P. Lescot, and M. Röckner, “The global random attractor for a class of stochastic porous media equations,” Communications in Partial Differential Equations, vol. 36, no. 3, pp. 446–469, 2011. MR2763333 10.1080/03605302.2010.523919

17.

Z. Wang, S. Zhou, and A. Gu, “Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains,” Nonlinear Analysis: Real World Applications, vol. 12, no. 6, pp. 3468–3482, 2011. MR2832985 1231.35025 10.1016/j.nonrwa.2011.06.008 Z. Wang, S. Zhou, and A. Gu, “Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains,” Nonlinear Analysis: Real World Applications, vol. 12, no. 6, pp. 3468–3482, 2011. MR2832985 1231.35025 10.1016/j.nonrwa.2011.06.008

18.

Z. Wang and S. Zhou, “Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains,” Journal of Mathematical Analysis and Applications, vol. 384, no. 1, pp. 160–172, 2011. MR2822858 1227.60087 10.1016/j.jmaa.2011.02.082 Z. Wang and S. Zhou, “Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains,” Journal of Mathematical Analysis and Applications, vol. 384, no. 1, pp. 160–172, 2011. MR2822858 1227.60087 10.1016/j.jmaa.2011.02.082

19.

W. Zhao and Y. Li, “(\emphL$^{2}$,\emphL$^{p}$)-random attractors for stochastic reaction-diffusion equation on unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 2, pp. 485–502, 2012. MR2847434 1229.60081 W. Zhao and Y. Li, “(\emphL$^{2}$,\emphL$^{p}$)-random attractors for stochastic reaction-diffusion equation on unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 2, pp. 485–502, 2012. MR2847434 1229.60081

20.

A. Gu, “Random attractors of stochastic three-component reversible Gray-Scott system on unbounded domains,” Abstract and Applied Analysis, vol. 2012, Article ID 419472, 22 pages, 2012. MR2965460 1254.37038 A. Gu, “Random attractors of stochastic three-component reversible Gray-Scott system on unbounded domains,” Abstract and Applied Analysis, vol. 2012, Article ID 419472, 22 pages, 2012. MR2965460 1254.37038

21.

L. Wang and D. Xu, “Asymptotic behavior of a class of reaction-diffusion equations with delays,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 439–453, 2003. MR1982665 1031.35065 10.1016/S0022-247X(03)00112-4 L. Wang and D. Xu, “Asymptotic behavior of a class of reaction-diffusion equations with delays,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 439–453, 2003. MR1982665 1031.35065 10.1016/S0022-247X(03)00112-4

22.

T. Caraballo and J. Real, “Attractors for 2D-Navier-Stokes models with delays,” Journal of Differential Equations, vol. 205, no. 2, pp. 271–297, 2004. MR2091818 1068.35088 10.1016/j.jde.2004.04.012 T. Caraballo and J. Real, “Attractors for 2D-Navier-Stokes models with delays,” Journal of Differential Equations, vol. 205, no. 2, pp. 271–297, 2004. MR2091818 1068.35088 10.1016/j.jde.2004.04.012

23.

J. Li and J. Huang, “Uniform attractors for non-autonomous parabolic equations with delays,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 2194–2209, 2009. MR2524429 1177.35241 J. Li and J. Huang, “Uniform attractors for non-autonomous parabolic equations with delays,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 2194–2209, 2009. MR2524429 1177.35241

24.

P. Marín-Rubio, J. Real, and J. Valero, “Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 5, pp. 2012–2030, 2011. MR2764398 1218.35045 P. Marín-Rubio, J. Real, and J. Valero, “Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 5, pp. 2012–2030, 2011. MR2764398 1218.35045

25.

M. Conti and M. Coti Zelati, “Attractors for the Cahn-Hilliard equation with memory in 2D,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1668–1682, 2010. MR2577567 1180.35118 M. Conti and M. Coti Zelati, “Attractors for the Cahn-Hilliard equation with memory in 2D,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1668–1682, 2010. MR2577567 1180.35118

26.

P. Marín-Rubio and J. Real, “Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 10, pp. 2784–2799, 2007. MR2347436 1131.35059 P. Marín-Rubio and J. Real, “Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 10, pp. 2784–2799, 2007. MR2347436 1131.35059

27.

I. Chueshov and M. Scheutzow, “On the structure of attractors and invariant measures for a class of monotone random systems,” Dynamical Systems, vol. 19, no. 2, pp. 127–144, 2004. MR2060422 1061.37032 10.1080/1468936042000207792 I. Chueshov and M. Scheutzow, “On the structure of attractors and invariant measures for a class of monotone random systems,” Dynamical Systems, vol. 19, no. 2, pp. 127–144, 2004. MR2060422 1061.37032 10.1080/1468936042000207792

28.

X. Ding and J. Jiang, “Random attractors for stochastic retarded lattice dynamical systems,” Abstract and Applied Analysis, vol. 2012, Article ID 409282, 27 pages, 2012. MR2984047 X. Ding and J. Jiang, “Random attractors for stochastic retarded lattice dynamical systems,” Abstract and Applied Analysis, vol. 2012, Article ID 409282, 27 pages, 2012. MR2984047

29.

W. Yan, Y. Li, and S. Ji, “Random attractors for first order stochastic retarded lattice dynamical systems,” Journal of Mathematical Physics, vol. 51, no. 3, Article ID 032702, 17 pages, 2010. MR2647873 W. Yan, Y. Li, and S. Ji, “Random attractors for first order stochastic retarded lattice dynamical systems,” Journal of Mathematical Physics, vol. 51, no. 3, Article ID 032702, 17 pages, 2010. MR2647873

30.

B. Wang, “Attractors for reaction-diffusion equations in unbounded domains,” Physica D, vol. 128, no. 1, pp. 41–52, 1999. MR1685247 0953.35022 10.1016/S0167-2789(98)00304-2 B. Wang, “Attractors for reaction-diffusion equations in unbounded domains,” Physica D, vol. 128, no. 1, pp. 41–52, 1999. MR1685247 0953.35022 10.1016/S0167-2789(98)00304-2

31.

A. Rodriguez-Bernal and B. Wang, “Attractors for partly dissipative reaction diffusion systems in ${\mathbb{R}}^{n}$,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 790–803, 2000. MR1800181 0977.35028 10.1006/jmaa.2000.7122 A. Rodriguez-Bernal and B. Wang, “Attractors for partly dissipative reaction diffusion systems in ${\mathbb{R}}^{n}$,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 790–803, 2000. MR1800181 0977.35028 10.1006/jmaa.2000.7122

32.

L. Arnold, Random Dynamical Systems, Springer, Berlin, Germany, 1998. MR1723992 1116.01326 L. Arnold, Random Dynamical Systems, Springer, Berlin, Germany, 1998. MR1723992 1116.01326

33.

I. Chueshov, Monotone Random Systems Theory and Applications, Springer, Berlin, Germany, 2002. MR1902500 I. Chueshov, Monotone Random Systems Theory and Applications, Springer, Berlin, Germany, 2002. MR1902500

34.

P. W. Bates, H. Lisei, and K. Lu, “Attractors for stochastic lattice dynamical systems,” Stochastics and Dynamics, vol. 6, no. 1, pp. 1–21, 2006. MR2210679 1105.60041 10.1142/S0219493706001621 P. W. Bates, H. Lisei, and K. Lu, “Attractors for stochastic lattice dynamical systems,” Stochastics and Dynamics, vol. 6, no. 1, pp. 1–21, 2006. MR2210679 1105.60041 10.1142/S0219493706001621

35.

G. da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992. MR1207136 G. da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992. MR1207136

36.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY, USA, 1983. MR710486 A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY, USA, 1983. MR710486

37.

A. Bensoussan, G. da Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäauser, Boston, Mass, USA, 2nd edition, 2007.  MR2273323 A. Bensoussan, G. da Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Birkhäauser, Boston, Mass, USA, 2nd edition, 2007.  MR2273323
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Xiaoquan Ding and Jifa Jiang "Random Attractors for Stochastic Retarded Reaction-Diffusion Equations on Unbounded Domains," Abstract and Applied Analysis 2013(none), 1-16, (2013). https://doi.org/10.1155/2013/981576
Published: 2013
Vol.2013 • 2013
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