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Spatial Homogenization of Stochastic Wave Equation with Large Interaction

Published online by Cambridge University Press:  20 November 2018

Yongxin Jiang ,
Wei Wang  and
Zhaosheng Feng
Yongxin Jiang
Affiliation:
Department of Mathematics, Hohai University, Nanjing, Jiangsu 210098, China e-mail: yxinjiang@hhu.edu.cn
Wei Wang
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, China e-mail: wangweinju@aliyun.com
Zhaosheng Feng
Affiliation:
Department of Mathematics, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA e-mail: zhaosheng.feng@utrgv.edu
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Abstract

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A dynamical approximation of a stochastic wave equation with large interaction is derived. A random invariant manifold is discussed. By a key linear transformation, the random invariant manifold is shown to be close to the random invariant manifold of a second-order stochastic ordinary differential equation.

Keywords

60F1060H1535Q55stochasticwave equationhomogeneous systemapproximationrandom invariant manifold, Neumann boundary condition
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 3 , 01 September 2016 , pp. 542 - 552
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Arnold, L., Random dynamical systems. Springer Monographs in Mather Springer-Verlag, Berlin, 1998. http://dx.doi.org/10.1007/978-3-662-1287! Google Scholar
[2] Carvalho, A. N. and Hale, J. K., Large diffusion with dispersion. Nonlineano. 12, 1139–1151. http://dx.doi.org/10.1016/0362-546X(91)90233-Q Google Scholar
[3] Cerrai, S. and Freidlin, M., On the Smoluchowski-Kramers approximation infinite number of degrees of freedom.Probab. Theory Related Fields 135 363–394. http://dx.doi.org/!0.1007/s00440-005-0465-0 Google Scholar
[4] Cerrai, S. and Freidlin, M., Smoluchowski-Kramers approximation for a general class ofSPDEs. J. Evol. Equ. 6(2006), no. 4, 657–689. http://dx.doi.org/10.1007/s00028-006-0281-8 Google Scholar
[5] Duan, J., Lu, K., and Schmalfuss, B., Invariant manifolds for stochastic partial differential equations. Ann. of Prob. 31(2003), no. 4, 2109–2135.http://dx.doi.org/10.1214/aop/1068646380 Google Scholar
[6] Duan, J., Lu, K., and Schmalfuss, B., Smooths stable and unstable manifolds for stochastic evolutionary equations. J. Dynam. Differential Equations 16(2004), no. 4, 949–972. http://dx.doi.org/10.1007/s10884- 004- 7830-z Google Scholar
[7] Liu, Z., Stochastic inertial manifolds for damped wave equations. Stoch.Dyn. 10(2010), no. 2, 211–230. http://dx.doi.org/10.1142/S021 949371 000292 9 Google Scholar
[8] Lu, K. and Schmalfuss, B., Invariant manifolds for stochastic wave equations. J. Differential Equations 236(2007), no. 2, 460–492. http://dx.doi.org/10.101 6/j.jde.2006.09.024 Google Scholar
[9] Lv, Y. and Roberts, A. J., Averaging approximation to singularly perturbed stochastic wave equation. J. Math. Phys. 53(2012), no. 6, 062702, 11.http://dx.doi.org/10.1 063/1.47261 75 Google Scholar
[10] Lv, Y. and Wang, W., Limit dynamics for stochastic wave equations. J. Differential Equations 244(2008), 1–23. http://dx.doi.org/10.1016/j.jde.2007.10.009 Google Scholar
[11] Lv, Y., Wang, W., and Roberts, A., Approximation of the random inertial manifold of singularly perturbed stochastic wave equations. Stoch.Dyn. 14(2014), no. 2,1350018, 21pp. http://dx.doi.org/10.1142/S021 949371 35001 84 Google Scholar
[12] Mora, X., Finite-dimensional attracting invariant manifolds for damped semilinear wave equations. In: Contributions to nonlinear partial differential equations. II. (Paris,1985), Pitman Res. Notes Math. Ser., 155, Longman Sci. Tech., Harlow, 1987, pp. 172–183.Google Scholar
[13] Pazy, A., Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. http://dx.doi.org/!0.1007/978-1-4612-5561-1 Google Scholar
[14] Da Prato, G. and Zabczyk, J., Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992. http://dx.doi.org/10.1017/CBO9780511666223 Google Scholar
[15] Qin, W.-X., Spatial homogeneity and invariant manifolds for damped hyperbolic equations. Z. Angew.Math. Phys. 52(2001), 990–1016. http://dx.doi.org/10.1007/PL00001591 Google Scholar
[16] Roberts, A. J., Resolving the multitude of microscale interactions accurately models stochastic partial differential equations. LMS J. Comput. and Math. 9(2006), 193–221. http://dx.doi.org/10.1112/S14611 5700000125X Google Scholar
[17] Roberts, A. J., Normal form transforms separate slow and fast modes in stochastic dynamical systems. 387(2008), no. 1,12-38. http://dx.doi.org/10.1016/j.physa.2007.08.023 Google Scholar
[18] Wang, W. and Lv, Y., Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete Contin.Dyn.Syst. Ser. B 13(2010), no. 1,175-193. http://dx.doi.org/! O.3934/dcdsb.2O1 0.13.1 75 Google Scholar