Journal ArticleDOI
Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise
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In this article, the authors studied the asymptotic behavior of the solutions of the fractional reaction-diffusion equations with polynomial drift terms of arbitrary order driven by locally Lipschitz nonlinear diffusion terms defined on R n.About:
This article is published in Journal of Differential Equations.The article was published on 2019-12-15. It has received 51 citations till now. The article focuses on the topics: Lipschitz continuity & Random dynamical system.read more
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Random dynamics of p-Laplacian lattice systems driven by infinite-dimensional nonlinear noise
Renhai Wang,Bixiang Wang +1 more
TL;DR: In this article, the existence and uniqueness of mean square solutions to the equations are proved when the nonlinear drift and diffusion terms are locally Lipschitz continuous and it is shown that the mean random dynamical system generated by the solution operators has a unique tempered weak pullback random attractor in a Bochner space.
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Periodic measures of stochastic delay lattice systems
TL;DR: The periodic measures of the stochastic delay reaction-diffusion lattice systems are investigated in this paper, and the existence of periodic measures when the time-dependent terms of the system are periodic in time is shown.
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Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence.
TL;DR: In this paper, a new approach to nonlinear stochastic partial differential equations with Gaussian noise is developed, which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type.
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Limiting Behavior of Invariant Measures of Stochastic Delay Lattice Systems
TL;DR: In this article, it was shown that any limit point of a tight sequence of invariant measures of the stochastic delay lattice systems must be an invariant measure of the corresponding limiting system as the intensity of noise converges or the time-delay approaches zero.
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Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on ℝN driven by nonlinear noise
TL;DR: In this article, the well-posedness of a wide class of non-autonomous, non-local, fractional, stochastic FitzHugh-Nagumo systems driven by nonlinear noise defined on the entire space is investigated.
References
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Book
Stochastic Equations in Infinite Dimensions
Giuseppe Da Prato,Jerzy Zabczyk +1 more
TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
Journal ArticleDOI
Hitchhiker's guide to the fractional Sobolev spaces
TL;DR: In this article, the authors deal with the fractional Sobolev spaces W s;p and analyze the relations among some of their possible denitions and their role in the trace theory.
Journal ArticleDOI
Attractors for random dynamical systems
Hans Crauel,Franco Flandoli +1 more
TL;DR: In this article, a criterion for existence of global random attractors for RDS is established and the existence of invariant Markov measures supported by the random attractor is proved for SPDE, which yields invariant measures for the associated Markov semigroup.
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The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary
Xavier Ros-Oton,Joaquim Serra +1 more
TL;DR: In this article, the Pohozaev identity up to the boundary of the Dirichlet problem for the fractional Laplacian was shown to hold for the case of ( − Δ ) s u = g in Ω, u ≡ 0 in R n \ Ω, for some s ∈ ( 0, 1 ) and g ∈ L ∞ ( Ω ), then u is C s ( R n ) and u / δ s | Ω is C α up to boundary ∂Ω for some α ∈( 0