# American Institute of Mathematical Sciences

April 2018, 38(4): 1983-2005. doi: 10.3934/dcds.2018080

## Large deviations for stochastic heat equations with memory driven by Lévy-type noise

 1 Department of Mathematics, King's College London, London WC2R 2LS, United Kingdom 2 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

* Corresponding author

Received  January 2017 Revised  October 2017 Published  January 2018

Fund Project: The second author acknowledges funding by K. C. Wong Foundation for a 1-year fellowship at King's College London

For a heat equation with memory driven by a Lévy-type noise we establish the existence of a unique solution. The main part of the article focuses on the Freidlin-Wentzell large deviation principle of the solutions of heat equation with memory driven by a Lévy-type noise. For this purpose, we exploit the recently introduced weak convergence approach.

Citation: Markus Riedle, Jianliang Zhai. Large deviations for stochastic heat equations with memory driven by Lévy-type noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1983-2005. doi: 10.3934/dcds.2018080
##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, New York, NY: Academic Press, 2003. [2] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory: Theory and Applications, New York: Springer, 2012. [3] J. Bao and C. Yuan, Large deviations for neutral functional SDEs with jumps, Stochastics, 87 (2015), 48-70. doi: 10.1080/17442508.2014.914516. [4] V. Barbu, S. Bonaccorsi and L. Tubaro, Existence and asymptotic behavior for hereditary stochastic evolution equations, Appl. Math. Optim., 69 (2014), 273-314. doi: 10.1007/s00245-013-9224-2. [5] S. Bonaccorsi, G. da Prato and L. Tubaro, Asymptotic behavior of a class of nonlinear stochastic heat equations with memory effects, SIAM J. Math. Anal., 44 (2012), 1562-1587. doi: 10.1137/110841795. [6] A. Budhiraja, J. Chen and P. Dupuis, Large deviations for stochastic partial differential equations driven by a Poisson random measure, Stochastic Processes Appl., 123 (2013), 523-560. doi: 10.1016/j.spa.2012.09.010. [7] A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincaré, Probab. Stat., 47 (2011), 725-747. doi: 10.1214/10-AIHP382. [8] 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. [9] 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. [10] B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208. doi: 10.1007/BF01596912. [11] Z. Dong, J. Xiong, J. Zhai and T. Zhang, A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, J. Funct. Anal., 272 (2017), 227-254. doi: 10.1016/j.jfa.2016.10.012. [12] C. Giorgi and V. Pata, Asymptotic behavior of a nonlinear hyperbolic heat equation with memory, NoDEA, Nonlinear Differ. Equ. Appl., 8 (2001), 157-171. doi: 10.1007/PL00001443. [13] C. Giorgi, V. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA, Nonlinear Differ. Equ. Appl., 5 (1998), 333-354. doi: 10.1007/s000300050049. [14] B. Goldys, M. Röckner and X. Zhang, Martingale solutions and Markov selections for stochastic partial differential equations, Stochastic Processes Appl., 119 (2009), 1725-1764. doi: 10.1016/j.spa.2008.08.009. [15] M. E. Gurtin and B. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373. [16] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Amsterdam: North-Holland Publishing Company, 1981. [17] Y. Li, Y. Xie and X. Zhang, Large deviation principle for stochastic heat equation with memory, Discrete Contin. Dyn. Syst., 35 (2015), 5221-5237. doi: 10.3934/dcds.2015.35.5221. [18] J. W. Nunziato, On heat conduction in materials with memory, Q. Appl. Math., 29 (1971), 187-204. doi: 10.1090/qam/295683. [19] J. Xiong and J. Zhai, Large deviations for locally monotone stochastic partial differential equations driven by levy noise, 2017. Available from: https://www.e-publications.org/ims/submission/BEJ/user/submissionFile/27769?confirm=1f6bdfb2. [20] X. Yang, J. Zhai and T. Zhang, Large deviations for SPDEs of jump type, Stoch. Dyn. , 15 (2015), 1550026, 30 pp. [21] J. Zhai and T. Zhang, Large deviations for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, Bernoulli, 21 (2015), 2351-2392. doi: 10.3150/14-BEJ647.

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##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, New York, NY: Academic Press, 2003. [2] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory: Theory and Applications, New York: Springer, 2012. [3] J. Bao and C. Yuan, Large deviations for neutral functional SDEs with jumps, Stochastics, 87 (2015), 48-70. doi: 10.1080/17442508.2014.914516. [4] V. Barbu, S. Bonaccorsi and L. Tubaro, Existence and asymptotic behavior for hereditary stochastic evolution equations, Appl. Math. Optim., 69 (2014), 273-314. doi: 10.1007/s00245-013-9224-2. [5] S. Bonaccorsi, G. da Prato and L. Tubaro, Asymptotic behavior of a class of nonlinear stochastic heat equations with memory effects, SIAM J. Math. Anal., 44 (2012), 1562-1587. doi: 10.1137/110841795. [6] A. Budhiraja, J. Chen and P. Dupuis, Large deviations for stochastic partial differential equations driven by a Poisson random measure, Stochastic Processes Appl., 123 (2013), 523-560. doi: 10.1016/j.spa.2012.09.010. [7] A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincaré, Probab. Stat., 47 (2011), 725-747. doi: 10.1214/10-AIHP382. [8] 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. [9] 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. [10] B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208. doi: 10.1007/BF01596912. [11] Z. Dong, J. Xiong, J. Zhai and T. Zhang, A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, J. Funct. Anal., 272 (2017), 227-254. doi: 10.1016/j.jfa.2016.10.012. [12] C. Giorgi and V. Pata, Asymptotic behavior of a nonlinear hyperbolic heat equation with memory, NoDEA, Nonlinear Differ. Equ. Appl., 8 (2001), 157-171. doi: 10.1007/PL00001443. [13] C. Giorgi, V. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA, Nonlinear Differ. Equ. Appl., 5 (1998), 333-354. doi: 10.1007/s000300050049. [14] B. Goldys, M. Röckner and X. Zhang, Martingale solutions and Markov selections for stochastic partial differential equations, Stochastic Processes Appl., 119 (2009), 1725-1764. doi: 10.1016/j.spa.2008.08.009. [15] M. E. Gurtin and B. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rat. Mech. Anal., 31 (1968), 113-126. doi: 10.1007/BF00281373. [16] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Amsterdam: North-Holland Publishing Company, 1981. [17] Y. Li, Y. Xie and X. Zhang, Large deviation principle for stochastic heat equation with memory, Discrete Contin. Dyn. Syst., 35 (2015), 5221-5237. doi: 10.3934/dcds.2015.35.5221. [18] J. W. Nunziato, On heat conduction in materials with memory, Q. Appl. Math., 29 (1971), 187-204. doi: 10.1090/qam/295683. [19] J. Xiong and J. Zhai, Large deviations for locally monotone stochastic partial differential equations driven by levy noise, 2017. Available from: https://www.e-publications.org/ims/submission/BEJ/user/submissionFile/27769?confirm=1f6bdfb2. [20] X. Yang, J. Zhai and T. Zhang, Large deviations for SPDEs of jump type, Stoch. Dyn. , 15 (2015), 1550026, 30 pp. [21] J. Zhai and T. Zhang, Large deviations for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises, Bernoulli, 21 (2015), 2351-2392. doi: 10.3150/14-BEJ647.
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