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Estimates on the distance of inertial manifolds
Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients
1. | University of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie, WY 82071, United States |
2. | Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain |
3. | Institut für Stochastik, Friedrich Schiller Universität Jena, Ernst Abbe Platz 2, 07737 Jena |
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. |
[2] |
P. R. Beesack, Gronwall Inequalities, Carleton Mathematical Lecture Notes, 1975. |
[3] |
A. Bensoussan and J. Frehse, Local solutions for stochastic Navier Stokes equations, Mathematical Modelling and Numerical Analysis (Modélisation Mathématique et Analyse Numérique) 34 (2000), 241-273.
doi: 10.1051/m2an:2000140. |
[4] |
T. Caraballo, M. J. Garrido-Atienza and J. Real, Existence and uniqueness of solutions for delay stochastic evolution equations, Stochastic Anal. Appl., 20 (2002), 1225-1256.
doi: 10.1081/SAP-120015831. |
[5] |
T. Caraballo, M. J. Garrido-Atienza and B. Schmalfuss, Existence of exponentially attracting stationary solutions for delay evolution equations, Disc. Contin. Dyn. Syst., 18 (2007), 271-293.
doi: 10.3934/dcds.2007.18.271. |
[6] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Disc. Contin. Dyn. Syst. Ser. A, 21 (2008), 415-443.
doi: 10.3934/dcds.2008.21.415. |
[7] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Asymptotic behavior of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Disc. Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.
doi: 10.3934/dcdsb.2010.14.439. |
[8] |
T. Caraballo, P. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optimization, 50 (2004), 183-207.
doi: 10.1007/s00245-004-0802-1. |
[9] |
T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 1775-1802.
doi: 10.1098/rspa.2000.0586. |
[10] |
T. Caraballo, J. Real and I. D. 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. |
[11] |
D. Cheban and B. Schmalfuß, Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations, J. Math. Anal. Appl., 340 (2008), 374-393.
doi: 10.1016/j.jmaa.2007.08.046. |
[12] |
Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs and random dynamical systems, to appear in Discrete Contin. Dyn. Syst. Ser. A..
doi: 10.3934/dcds.2014.34.79. |
[13] |
Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Random attractors for SPDEs driven by an fBm, submitted. |
[14] |
I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b83277. |
[15] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223. |
[16] |
J. A. Dieudonné, Foundations of Modern Analysis, New York,Academic Press, 1964. |
[17] |
J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations, Ann. Prob., 31 (2003), 2109-2135.
doi: 10.1214/aop/1068646380. |
[18] |
J. Duan, K. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.
doi: 10.1007/s10884-004-7830-z. |
[19] |
F. Flandoli and H. Lisei, Stationary conjugation of flows for parabolic SPDEs with multiplicative noise and some applications, Stochastic Anal. Appl., 22 (2004), 1385-1420.
doi: 10.1081/SAP-200029481. |
[20] |
F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochast. Rep., 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[21] |
M. J. Garrido-Atienza and J. Real, Existence and uniqueness of solutions for delay stochastic evolution equations of second order in time, Stoch. Dyn., 3 (2003), 141-167.
doi: 10.1142/S0219493703000723. |
[22] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1995.
doi: 10.1007/978-1-4612-4342-7. |
[23] |
P. Imkeller and B. Schmalfuß, The conjugacy of stochastic and random differential equations and the existence of global attractors, J. Dynam. Differential Equations, 13 (2001), 215-249.
doi: 10.1023/A:1016673307045. |
[24] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993. |
[25] |
H. Kunita, Stochastic Flows and Stochastic Differential Equations, University Press, Cambridge, 1990. |
[26] | |
[27] |
X. Mao, Stability of Stochastic Differential Equations with Respect to Semimartingales, Pitman Research Notes in Mathematics Series, 251. Longman Scientific & Technical, Harlow, 1991. |
[28] |
X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. |
[29] |
W. Mather, M. R. Bennett, J. Hasty and L. S. Tsimring, Delay-induced degrade- and-fire oscillations in small genetics circuits, Phys. Rev. Lett., 102 (2009), 1-4.
doi: 10.1103/PhysRevLett.102.068105. |
[30] |
S.-E. A. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, Vol. 99, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1984. |
[31] |
S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for non-linear stochastic systems with memory. I. Existence of the semiflow, Journal of Functional Analysis, 205 (2003), 271-305.
doi: 10.1016/j.jfa.2002.04.001. |
[32] |
J. D. Murray, Mathematical Biology, Springer, 1993.
doi: 10.1007/b98869. |
[33] |
J. Real, Stochastic partial differential equations with delays, Stochastics, 8 (1982-83), 81-102.
doi: 10.1080/17442508208833230. |
[34] |
B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, in Int. Seminar on Applied Mathematics Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), 1992, 185-192. |
[35] |
B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in Int. Conf. Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 684-689. |
[36] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. |
[37] |
T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, Journal of Differential Equations, 181 (2002), 72-91.
doi: 10.1006/jdeq.2001.4073. |
[38] |
K. Yoshida, Functional Analysis, Sixth edition, Grundlehren der mathematischen Wissenschaften, 123, Springer-Verlag, Berlin-New York, 1980. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. |
[2] |
P. R. Beesack, Gronwall Inequalities, Carleton Mathematical Lecture Notes, 1975. |
[3] |
A. Bensoussan and J. Frehse, Local solutions for stochastic Navier Stokes equations, Mathematical Modelling and Numerical Analysis (Modélisation Mathématique et Analyse Numérique) 34 (2000), 241-273.
doi: 10.1051/m2an:2000140. |
[4] |
T. Caraballo, M. J. Garrido-Atienza and J. Real, Existence and uniqueness of solutions for delay stochastic evolution equations, Stochastic Anal. Appl., 20 (2002), 1225-1256.
doi: 10.1081/SAP-120015831. |
[5] |
T. Caraballo, M. J. Garrido-Atienza and B. Schmalfuss, Existence of exponentially attracting stationary solutions for delay evolution equations, Disc. Contin. Dyn. Syst., 18 (2007), 271-293.
doi: 10.3934/dcds.2007.18.271. |
[6] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Disc. Contin. Dyn. Syst. Ser. A, 21 (2008), 415-443.
doi: 10.3934/dcds.2008.21.415. |
[7] |
T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Asymptotic behavior of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Disc. Contin. Dyn. Syst. Ser. B, 14 (2010), 439-455.
doi: 10.3934/dcdsb.2010.14.439. |
[8] |
T. Caraballo, P. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optimization, 50 (2004), 183-207.
doi: 10.1007/s00245-004-0802-1. |
[9] |
T. Caraballo, K. Liu and A. Truman, Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 1775-1802.
doi: 10.1098/rspa.2000.0586. |
[10] |
T. Caraballo, J. Real and I. D. 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. |
[11] |
D. Cheban and B. Schmalfuß, Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations, J. Math. Anal. Appl., 340 (2008), 374-393.
doi: 10.1016/j.jmaa.2007.08.046. |
[12] |
Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs and random dynamical systems, to appear in Discrete Contin. Dyn. Syst. Ser. A..
doi: 10.3934/dcds.2014.34.79. |
[13] |
Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Random attractors for SPDEs driven by an fBm, submitted. |
[14] |
I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b83277. |
[15] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223. |
[16] |
J. A. Dieudonné, Foundations of Modern Analysis, New York,Academic Press, 1964. |
[17] |
J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations, Ann. Prob., 31 (2003), 2109-2135.
doi: 10.1214/aop/1068646380. |
[18] |
J. Duan, K. Lu and B. Schmalfuß, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.
doi: 10.1007/s10884-004-7830-z. |
[19] |
F. Flandoli and H. Lisei, Stationary conjugation of flows for parabolic SPDEs with multiplicative noise and some applications, Stochastic Anal. Appl., 22 (2004), 1385-1420.
doi: 10.1081/SAP-200029481. |
[20] |
F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochast. Rep., 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[21] |
M. J. Garrido-Atienza and J. Real, Existence and uniqueness of solutions for delay stochastic evolution equations of second order in time, Stoch. Dyn., 3 (2003), 141-167.
doi: 10.1142/S0219493703000723. |
[22] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1995.
doi: 10.1007/978-1-4612-4342-7. |
[23] |
P. Imkeller and B. Schmalfuß, The conjugacy of stochastic and random differential equations and the existence of global attractors, J. Dynam. Differential Equations, 13 (2001), 215-249.
doi: 10.1023/A:1016673307045. |
[24] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993. |
[25] |
H. Kunita, Stochastic Flows and Stochastic Differential Equations, University Press, Cambridge, 1990. |
[26] | |
[27] |
X. Mao, Stability of Stochastic Differential Equations with Respect to Semimartingales, Pitman Research Notes in Mathematics Series, 251. Longman Scientific & Technical, Harlow, 1991. |
[28] |
X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. |
[29] |
W. Mather, M. R. Bennett, J. Hasty and L. S. Tsimring, Delay-induced degrade- and-fire oscillations in small genetics circuits, Phys. Rev. Lett., 102 (2009), 1-4.
doi: 10.1103/PhysRevLett.102.068105. |
[30] |
S.-E. A. Mohammed, Stochastic Functional Differential Equations, Research Notes in Mathematics, Vol. 99, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1984. |
[31] |
S.-E. A. Mohammed and M. K. R. Scheutzow, The stable manifold theorem for non-linear stochastic systems with memory. I. Existence of the semiflow, Journal of Functional Analysis, 205 (2003), 271-305.
doi: 10.1016/j.jfa.2002.04.001. |
[32] |
J. D. Murray, Mathematical Biology, Springer, 1993.
doi: 10.1007/b98869. |
[33] |
J. Real, Stochastic partial differential equations with delays, Stochastics, 8 (1982-83), 81-102.
doi: 10.1080/17442508208833230. |
[34] |
B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, in Int. Seminar on Applied Mathematics Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), 1992, 185-192. |
[35] |
B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in Int. Conf. Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 684-689. |
[36] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. |
[37] |
T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, Journal of Differential Equations, 181 (2002), 72-91.
doi: 10.1006/jdeq.2001.4073. |
[38] |
K. Yoshida, Functional Analysis, Sixth edition, Grundlehren der mathematischen Wissenschaften, 123, Springer-Verlag, Berlin-New York, 1980. |
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