On random attractor of semilinear stochastically perturbed wave equation without uniqueness
Abstract
In this paper we investigate the dynamics of solutions of the semilinear wave equation, perturbed by additive white noise, in sense of the random attractor theory. The conditions on the parameters of the problem do not guarantee uniqueness of solution of the corresponding Cauchy problem. We prove theorem on the existence of random attractor for abstract noncompact multi-valued random dynamical system, which is applied to the wave equation with non-smooth nonlinear term. A priory estimate for weak solution of randomly perturbed problem is deduced, which allows to obtain the existence at least one weak solution. The multi-valued stochastic flow is generated by the weak solutions of investigated problem. We prove the existence of random attractor for generated multi-valued stochastic flow.References
Crauel H., Flandoli F. Attractors for random dynamical systems // Prob. Theory and Related Fields. — 1994. — № 100. — P. 365–393.
Crauel H., Debussche A., Flandoli F. Random attractors // J. Dyn. Diff. Eqns. — 1997. — № 9. — P. 307–341.
Crauel H. Global random attractors are uniquely determined by attracting deterministic compact sets // Ann. Mat. Pura Appl. — 1999. — № 126. — P. 57–72.
Arnold L. Random dynamical systems. — NY: Springer, 1998. — 586 p.
Chueshov I. Monotone random systems, theory and applications. — NY: Springer, 2002. — 234 p.
Fan X. Random attractors for a damped sine-Gordon equation with white noise // Pacific J. Math. — 2004. — № 216. — P. 63–76.
Zhou S., Yin F., Ouyang Z. Random attractor for damped nonlinear wave equation with white noise // SIADS. — 2005. — № 4. — P. 883–903.
Wang B. Asymptotic behavior of stochastic wave equation with critical exponents on R3 // Trans. Amer. Math. Soc. — 2011. — № 363. — P. 3639–3663.
Mel’nik V.S. Multi-valued dynamics of nonlinear infinite-dimensional dynamical systems [in Russian]. — Kyiv, 1994. (Preprint / V.M. Glushkov Institute of Cybernetics of NASU; № 94–17).
Zgurovsky M.Z., Kasyanov P.O., Kapustyan O.V., Valero J., Zadoianchuk N.V. Evolution inclusions and variation inequalities for Earth data processing III. Long-time behavior of evolution inclusions solutions in Earth data analysis. — NY: Springer, 2012. — 330 p.
Caraballo T., Langa J., Valero J. Global attractors for multivalued random dynamical systems // Nonlinear Analysis. — 2002. — № 48. — P. 805–829.
Caraballo T., Langa J., Melnik V., Valero J. Pullback attractors of nonautonomous and stochastic multivalued dynamical systems // Set-Valued Analysis. — 2003. — № 11. — P. 153–201.
Kapustyan O.V. Random attractor of stochastically perturbed evolution inclusion [in Russian] // Differential Equations. — 2004. — № 40. — P. 1314–1318.
Kapustyan O.V. Random attractors for non-unique resolved dissipative in probability systems // Ukrain. Math. Journal. — 2004. — № 56. — P. 892–900.
Kapustyan O.V., Valero J., Pereguda O.V. Random attractors for the reaction-diffusion equation perturbed by a stochastic cadlag process // Theor. Probability and Math. Statist. — 2006. — № 73. — P. 57–69.
Iovane G., Kapustyan O.V. Random dynamics of stochastically perturbed evolution inclusion and problem of distribution of power in hierarchical structure // Journal of Automation and Information Sciences. — 2006. — № 4. — P. 122–135.
Ball J.M. Global attractors for damped semilinear wave equations // Discrete and Continuous Dynamical Systems. — 2004. — № 10. — P. 31–52.
Iovane G., Kapustyan O.V. Global attractors for non-autonomous wave equation without uniqueness of solution // System Research and Information Technologies. — 2006. — № 2. — P. 107–120.
