# American Institute of Mathematical Sciences

November  2015, 35(11): 5221-5237. doi: 10.3934/dcds.2015.35.5221

## Large deviation principle for stochastic heat equation with memory

 1 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, China, China 2 School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, China

Received  September 2013 Revised  March 2014 Published  May 2015

In this work, using the weak convergence argument, we prove a Freidlin-Wentzell's large deviation principle for a class of stochastic heat equations with memory and Dirichlet boundary conditions, where the nonlinear term is allowed to be of polynomial growth.
Citation: Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221
##### References:
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##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, 2003. [2] M. Boué and P. Dupuis, A variational representation for certain functionals of Brownian motion, Ann. of Prob., 26 (1998), 1641-1659. doi: 10.1214/aop/1022855876. [3] A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist., 20 (2000), 39-61. [4] A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. of Prob., 36 (2008), 1390-1420. doi: 10.1214/07-AOP362. [5] P. Cannarsa and D. Sforza, Global solutions of abstract semilinear parabolic equations with memory terms, Nonlin. Diff. Eqs. Appl., 10 (2003), 399-430. doi: 10.1007/s00030-003-1004-2. [6] T. Caraballo, I. D. Chueshov and J. Real, Pullback attractors for stochastic heat equation in materials with memory, Discrete and Continuous Dynamical Systems - Series B, 9 (2005), 525-539. doi: 10.3934/dcdsb.2008.9.525. [7] T. Caraballo, I. D. Chueshov, P. Marín-Rubio and J. Real, Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory, Discrete Contin. Dyn. Syst., 18 (2007), 253-270. doi: 10.3934/dcds.2007.18.253. [8] P. Clement and G. Da Prato, White noise perturbation of the heat equations in materials with memory, Dyn. Syst. Appl., 6 (1997), 441-460. [9] P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley, New-York, 1997. doi: 10.1002/9781118165904. [10] M. I. Freidlin and A. D. Wentzell, On small random perturbations of dynamical system, Russian Math. Surveys, 25 (1970), 1-55. [11] C. Giorgi, V. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 333-354. doi: 10.1007/s000300050049. [12] B. Goldys, M. Röckner and X. Zhang, Martingale solutions and Markov selections for stochastic evolution equations, Stoch. Proc. and Appl., 119 (2009), 1725-1764. doi: 10.1016/j.spa.2008.08.009. [13] O. Kallenberg, Foundations of Modern Probability, Second edition, Springer-Verlag, New-York, 2002. doi: 10.1007/978-1-4757-4015-8. [14] W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Appl. Math. Optim., 61 (2010), 27-56. doi: 10.1007/s00245-009-9072-2. [15] J. Ren and X. Zhang, Schilder theorem for the Brownian motion on the diffeomorphism group of the circle, J. Func. Anal., 224 (2005), 107-133. doi: 10.1016/j.jfa.2004.08.006. [16] J. Ren and X. Zhang, Freidlin-Wentzell's large deviations for stochastic evolution equations, J. Func. Anal., 254 (2008), 3148-3172. doi: 10.1016/j.jfa.2008.02.010. [17] M. Röckner, B. Schmuland and X. Zhang, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Condensed Matter Physics, 11 (2008), 247-259. doi: 10.5488/CMP.11.2.247. [18] D. W. Stroock, An Introduction to the Theory of Large Deviations, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4613-8514-1. [19] X. Zhang, A variational representation for random functionals on abstract Wiener spaces, J. Math. Kyoto Univ., 49 (2009), 475-490. [20] X. Zhang, On stochastic evolution equations with non-Lipschitz coefficients, Stochastic and Dynamics, 9 (2009), 549-595. doi: 10.1142/S0219493709002774.
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