Some properties of non-linear fractional stochastic heat equations on bounded domains. (September 2017)

• Record Type: Journal Article
• Authors: Foondun, Mohammud; Guerngar, Ngartelbaye; Nane, Erkan
• Is Part Of: Chaos, solitons and fractals. Volume 102(2017)
• Page Start: 86
• Page End: 93
• Publication Date: 2017-09
• Subjects: Stochastic fractional PDEs -- Large time behavior -- Colored noise; Primary 60H15; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2017.03.064
• Languages: English
• ISSNs: 0960-0779

Abstract

Abstract: We consider the following fractional stochastic partial differential equation on a bounded, open subset B of R d for d ≥ 1 ∂ t u t ( x ) = L u t ( x ) + ξ σ ( u t ( x ) ) F ˙ ( t, x ), where ξ is a positive parameter and σ is a globally Lipschitz continuous function. The stochastic forcing term F ˙ ( t, x ) is white in time but possibly colored in space. The operator L is fractional Laplacian which is the infinitesimal generator of a symmetric α -stable Lévy process in R d . We study the behaviour of the solution with respect to the parameter ξ .We show that under zero exterior boundary conditions, in the long run, the p th-moment of the solution grows exponentially fast for large values of ξ . However when ξ is very small we observe eventually an exponential decay of the p th-moment of this same solution.