# American Institute of Mathematical Sciences

October  2014, 34(10): 3945-3968. doi: 10.3934/dcds.2014.34.3945

## 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

Received  February 2013 Revised  May 2013 Published  April 2014

In this paper we study the long--time dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. For this purpose, we begin by showing the existence and uniqueness of a cocycle solution of such an equation. We do not assume that the noise is given in additive form or that it is a very simple multiplicative noise. However, we need some smoothing property for the coefficient in front of the noise. The main idea of this paper consists of expressing the stochastic integral in terms of non-stochastic integrals and the noisy path by using an integration by parts. This latter term causes that at first, only a local mild solution can be obtained, since in order to apply the Banach fixed point theorem it is crucial to have the Hölder norm of the noisy path to be sufficiently small. Subsequently, by using appropriate stopping times, we shall derive the existence and uniqueness of a global mild solution. Furthermore, the asymptotic behavior is investigated by using the Random Dynamical Systems theory. In particular, we shall show that the global mild solution generates a random dynamical system that, under an appropriate smallness condition for the time lag, has an associated random attractor.
Citation: Hakima Bessaih, María J. Garrido–Atienza, Björn Schmalfuss. Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 3945-3968. doi: 10.3934/dcds.2014.34.3945
##### 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] K. Liu, Personal Communication, (2013). [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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##### 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] K. Liu, Personal Communication, (2013). [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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