# American Institute of Mathematical Sciences

June  2008, 1(2): 353-363. doi: 10.3934/dcdss.2008.1.353

## Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise

 1 Southern Illinois University, Department of Mathematics, MC 4408, 1245 Lincoln Drive, Carbondale, IL 62901-7316

Received  September 2006 Revised  September 2007 Published  March 2008

One-dimensional wave equations with cubic power law perturbed by Q-regular additive space-time random noise are considered. These models describe the displacement of nonlinear strings excited by state-independent random external forces. The presented analysis is based on the representation of its solution in form of Fourier-series expansions along the eigenfunctions of Laplace operator with continuous, Markovian, unique Fourier coefficients (the so-called commutative case). We shall discuss existence and uniqueness of Fourier solutions using energy-type methods based on the construction of Lyapunov-functions. Appropriate truncations and finite-dimensional approximations are presented while exploiting the explicit knowledge on eigenfunctions of related second order differential operators. Moreover, some nonstandard partial-implicit difference methods for their numerical integration are suggested in order to control its energy functional in a dynamically consistent fashion. The generalized energy $\cE$ (sum of kinetic, potential and damping energy) is governed by the linear relation $\E [\varepsilon(t)] = \E [\varepsilon(0)] + b^2 trace (Q) t / 2$ in time $t \ge 0$, where $b$ is the scalar intensity of noise and $Q$ its covariance operator.
Citation: Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353
 [1] Henri Schurz. Stochastic heat equations with cubic nonlinearity and additive space-time noise in 2D. Conference Publications, 2013, 2013 (special) : 673-684. doi: 10.3934/proc.2013.2013.673 [2] Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607 [3] Ying Hu, Shanjian Tang. Nonlinear backward stochastic evolutionary equations driven by a space-time white noise. Mathematical Control & Related Fields, 2018, 8 (3&4) : 739-751. doi: 10.3934/mcrf.2018032 [4] Henri Schurz. Stochastic wave equations with cubic nonlinearity and Q-regular additive noise in $\mathbb{R}^2$. Conference Publications, 2011, 2011 (Special) : 1299-1308. doi: 10.3934/proc.2011.2011.1299 [5] Boris P. Belinskiy, Peter Caithamer. Energy of an elastic mechanical system driven by Gaussian noise white in time. Conference Publications, 2001, 2001 (Special) : 39-49. doi: 10.3934/proc.2001.2001.39 [6] Tianlong Shen, Jianhua Huang, Caibin Zeng. Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1523-1533. doi: 10.3934/dcdsb.2018056 [7] Angelo Favini, Georgy A. Sviridyuk, Alyona A. Zamyshlyaeva. One Class of Sobolev Type Equations of Higher Order with Additive "White Noise". Communications on Pure & Applied Analysis, 2016, 15 (1) : 185-196. doi: 10.3934/cpaa.2016.15.185 [8] Boling Guo, Guoli Zhou. On the backward uniqueness of the stochastic primitive equations with additive noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3157-3174. doi: 10.3934/dcdsb.2018305 [9] Georgios T. Kossioris, Georgios E. Zouraris. Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1845-1872. doi: 10.3934/dcdsb.2013.18.1845 [10] Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210 [11] Luis J. Roman, Marcus Sarkis. Stochastic Galerkin method for elliptic spdes: A white noise approach. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 941-955. doi: 10.3934/dcdsb.2006.6.941 [12] María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 [13] Xiaojun Li, Xiliang Li, Kening Lu. Random attractors for stochastic parabolic equations with additive noise in weighted spaces. Communications on Pure & Applied Analysis, 2018, 17 (3) : 729-749. doi: 10.3934/cpaa.2018038 [14] Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92 [15] Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221 [16] Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887 [17] T. Tachim Medjo. The exponential behavior of the stochastic primitive equations in two dimensional space with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 177-197. doi: 10.3934/dcdsb.2010.14.177 [18] Nathan Glatt-Holtz, Mohammed Ziane. The stochastic primitive equations in two space dimensions with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 801-822. doi: 10.3934/dcdsb.2008.10.801 [19] Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269 [20] Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381

2018 Impact Factor: 0.545