# zbMATH — the first resource for mathematics

Averaging principle for multiscale stochastic Klein-Gordon-heat system. (English) Zbl 1436.60063
Summary: This paper investigates multiscale stochastic Klein-Gordon-heat system. We establish the well-posedness and two kinds of stochastic averaging principle for stochastic Klein-Gordon-heat system with two timescales. To be more precise, under suitable conditions, two kinds of averaging principle (the autonomous case and the nonautonomous case) are proved, and as a consequence, the multiscale stochastic Klein-Gordon-heat system can be reduced to a single stochastic Klein-Gordon equation (averaged equation) with a modified coefficient, the slow component of multiscale stochastic system toward the solution of the averaged equation in moment (the autonomous case) and in probability (the nonautonomous case).

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 70K65 Averaging of perturbations for nonlinear problems in mechanics 70K70 Systems with slow and fast motions for nonlinear problems in mechanics 37H10 Generation, random and stochastic difference and differential equations 37L55 Infinite-dimensional random dynamical systems; stochastic equations
Full Text:
##### References:
 [1] Bréhier, CE, Strong and weak orders in averaging for SPDEs, Stoch. Process. Appl., 122, 2553-2593, (2012) · Zbl 1266.60112 [2] Bensoussan, A., Stochastic Navier-Stokes equations, Acta Appl. Math., 38, 267-304, (1995) · Zbl 0836.35115 [3] Bao, J.; Yin, G.; Yuan, C., Two-time-scale stochastic partial differential equations driven by $$\alpha$$-stable noises: averaging principles, Bernoulli, 23, 645-669, (2017) · Zbl 1360.60118 [4] Cerrai, S., A Khasminkii type averaging principle for stochastic reaction – diffusion equations, Ann. Appl. Probab., 19, 899-948, (2009) · Zbl 1191.60076 [5] Cerrai, S., Averaging principle for systems of reaction – diffusion equations with polynomial nonlinearities perturbed by multiplicative noise, SIAM J. Math. Anal., 43, 2482-2518, (2011) · Zbl 1239.60055 [6] Cerrai, S.; Freidlin, MI, Averaging principle for a class of stochastic reaction diffusion equations, Probab. Theory Relat. Fields, 144, 137-177, (2009) · Zbl 1176.60049 [7] Cerrai, S.; Lunardi, A., Averaging principle for nonautonomous slow – fast systems of stochastic reaction – diffusion equations: the almost periodic case, SIAM J. Math. Anal., 49, 2843-2884, (2017) · Zbl 1370.60102 [8] Chow, PL, Thermoelastic wave propagation in a random medium and some related problems, Int. J. Eng. Sci., 11, 953-971, (1973) · Zbl 0263.73007 [9] Chow, P.L.: Stochastic Partial Differential Equations. CRC Press, Boca Raton (2014) [10] Cardetti, F.; Choi, YS, A parabolic – hyperbolic system modelling a moving cell, Electron. J. Differ. Equ., 95, 1-11, (2009) · Zbl 1178.35389 [11] Choi, Y.; Miller, C., Global existence of solutions to a coupled parabolic – hyperbolic system with moving boundary, Proc. Am. Math. Soc., 139, 3257-3270, (2011) · Zbl 1232.35197 [12] Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (2014) · Zbl 1317.60077 [13] Dong, Z.; Sun, X.; Xiao, H.; Zhai, J., Averaging principle for one dimensional stochastic Burgers equation, J. Differ. Equ., 10, 4749-4797, (2018) · Zbl 1428.34061 [14] Dodd, R.K., Eilbeck, J.C., Gibbon, J.D., Morris, H.C.: Solitons and Nonlinear Wave Equations. Academic Press Inc, London (1982) · Zbl 0496.35001 [15] Debussche, A.; Glatt-Holtz, N.; Temam, R., Local martingale and pathwise solutions for an abstract fluids model, Physica D Nonlinear Phenom., 240, 1123-1144, (2011) · Zbl 1230.60065 [16] Fu, H.; Duan, J., An averaging principle for two-scale stochastic partial differential equations, Stoch. Dyn., 11, 353-367, (2011) · Zbl 1243.60053 [17] Fu, H.; Liu, J., Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations, J. Math. Anal. Appl., 384, 70-86, (2011) · Zbl 1223.60044 [18] Fu, H.; Wan, L.; Wang, Y.; Liu, J., Strong convergence rate in averaging principle for stochastic FitzHugh-Nagumo system with two time-scales, J. Math. Anal. Appl., 416, 609-628, (2014) · Zbl 1325.60107 [19] Fu, H.; Wan, L.; Liu, J., Strong convergence in averaging principle for stochastic hyperbolic – parabolic equations with two time-scales, Stoch. Process. Appl., 125, 3255-3279, (2015) · Zbl 1322.60111 [20] Fu, H.; Wan, L.; Liu, J.; etal., Weak order in averaging principle for stochastic wave equation with a fast oscillation, Stoch. Process. Appl., 128, 2557-2580, (2018) · Zbl 1396.60074 [21] Flato, M.; Simon, JCH; Taflin, E., Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations, Mem. Am. Math. Soc., 127, 311, (1997) · Zbl 0892.35147 [22] Gao, P., Global Carleman estimates for linear stochastic Kawahara equation and their applications, Math. Control Signals Syst., 28, 21, (2016) · Zbl 1355.35162 [23] Gao, P., Some periodic type solutions for stochastic reaction – diffusion equation with cubic nonlinearities, Comput. Math. Appl., 74, 2281-2297, (2017) · Zbl 1394.35590 [24] Gao, P., The stochastic Swift-Hohenberg equation, Nonlinearity, 30, 3516-3559, (2017) · Zbl 1372.60089 [25] Gao, P.: The solutions with recurrence property for stochastic linearly coupled complex cubic-quintic Ginzburg-Landau equations. Stoch. Dyn. (2018a). https://doi.org/10.1142/S0219493719500059 [26] Gao, P., Averaging principle for the higher order nonlinear Schrödinger equation with a random fast oscillation, J. Stat. Phys., 171, 897-926, (2018) · Zbl 1394.35473 [27] Gao, P., Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation, Discrete Contin. Dyn. Syst. A, 38, 5649-5684, (2018) · Zbl 1401.60121 [28] Gyöngy, I.; Krylov, N., Existence of strong solutions for Itô’s stochastic equations via approximations, Probab. Theory Relat. fields, 105, 143-158, (1996) · Zbl 0847.60038 [29] Khasminskii, RZ, On the principle of averaging the Itô stochastic differential equations, Kibernetika, 4, 260-279, (1968) [30] Leung, AW, Asymptotically stable invariant manifold for coupled nonlinear parabolic – hyperbolic partial differential equations, J. Differ. Equ., 187, 184-200, (2003) · Zbl 1022.35006 [31] Leung, AW, Stable invariant manifolds for coupled Navier-Stokes and second-order wave systems, Asymptot. Anal., 43, 339-357, (2005) · Zbl 1083.35010 [32] Lisei, H.; Keller, D., A stochastic nonlinear Schrödinger problem in variational formulation, Nonlinear Differ. Equ. Appl. NoDEA, 23, 1-27, (2016) · Zbl 1338.60160 [33] Lindblad, H.; Rodnianski, I., The weak null condition for Einsteins equations, C. R. Math. Acad. Sci. Paris, 336, 901-906, (2003) · Zbl 1045.35101 [34] Muñoz Rivera, JE; Racke, R., Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type, SIAM J. Math. Anal., 26, 1547-1563, (1995) · Zbl 0842.35039 [35] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1985) · Zbl 0516.47023 [36] Pei, B.; Xu, Y.; Wu, JL, Two-time-scales hyperbolic – parabolic equations driven by Poisson random measures: existence, uniqueness and averaging principles, J. Math. Anal. Appl., 447, 243-268, (2017) · Zbl 1387.60102 [37] Wu, S.; Chen, H.; Li, W., The local and global existence of the solutions of hyperbolic – parabolic system modeling biological phenomena, Acta Math. Sci., 28, 101-116, (2008) · Zbl 1150.35016 [38] Wang, W.; Roberts, AJ, Average and deviation for slow-fast stochastic partial differential equations, J. Differ. Equ., 253, 1265-1286, (2012) · Zbl 1251.35201 [39] Xu, J., $$L^p$$-strong convergence of the averaging principle for slow – fast SPDEs with jumps, J. Math. Anal. Appl., 445, 342-373, (2017) · Zbl 1348.60095 [40] Xu, J.; Miao, Y.; Liu, J., Strong averaging principle for two-time-scale non-autonomous stochastic FitzHugh-Nagumo system with jumps, J. Math. Phys., 57, 092704, (2016) · Zbl 1366.60088 [41] Yang, D.; Hou, Z., Large deviations for the stochastic derivative Ginzburg-Landau equation with multiplicative noise, Physica D Nonlinear Phenom., 237, 82-91, (2008) · Zbl 1172.60018 [42] Zhang, X.; Zuazua, E., Long-time behavior of a coupled heat-wave system arising in fluid – structure interaction, Arch. Ration. Mech. Anal., 184, 49, (2007) · Zbl 1178.74075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.