Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities

Caraballo, Tomás; Morillas, Francisco; Valero, José

doi:10.1016/j.jde.2012.03.020

Cited by 112 publications

(40 citation statements)

References 54 publications

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“…For stochastic lattice dynamical systems with additive or multiplicative noise, the existence of global random attractors has been intensively analyzed in the recent literature (see e.g., Bates et al [16], Caraballo et al [17,18], Caraballo and Lu [19], Han [20], Han et al [21], amongst others). We emphasize that in the studies of stochastic lattice systems with multiplicative noise up to date, only a finite number of Wiener process is considered in each equation, being the same in all the equations, while the multiplicative noise considered here in Equation (1.1) is different at each node.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

Caraballo

Han

Schmalfuß³

et al. 2016

Nonlinear Analysis

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In this paper we investigate the long term behavior of a stochastic lattice dynamical system with a diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and a different multiplicative white noise at each node. We prove that this stochastic lattice equation generates a random dynamical system that possesses a global random attractor. In particular, we first establish an existence theorem for weak solutions to general random evolution equations, which is later applied to the specific stochastic lattice system to show that it has weak solutions and the solutions generate a random dynamical system. We then prove the existence of a random attractor of the underlying random dynamical system by constructing a random compact absorbing set and using an embedding theorem. The major novelty of this work is that we consider a different multiplicative white noise term at each different node, which significantly improves the previous results in the literature where the same multiplicative noise was considered at all the nodes. As a consequence, the techniques used in the existing literature are not applicable here and a new methodology has to be developed to study such systems.

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Section: Introductionmentioning

confidence: 99%

“…This has subsequently become a standard way of formalization (see e.g. [19,18,21]). However, due to the appearance of the infinitely many noise terms, this scheme cannot be applied to handle our problem and hence a new methodology is ought to be developed.…”

Section: Introductionmentioning

confidence: 99%

Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

Caraballo

Han

Schmalfuß³

et al. 2016

Nonlinear Analysis

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“…Usually, the models under consideration are obtained by a spatial discretization of a parabolic or a hyperbolic equation (see e.g. [1], [2], [4], [5], [8], [11] [12], [15], [16], [19], [20], [22], [23], [26], [28], [29]). …”

Section: Introductionmentioning

confidence: 99%

Attractors for non-autonomous retarded lattice dynamical systems

Caraballo

Morillas

Valero

2015

Nonautonomous Dynamical Systems

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Abstract:In this paper we study a non-autonomous lattice dynamical system with delay. Under rather general growth and dissipative conditions on the nonlinear term, we de ne a non-autonomous dynamical system and prove the existence of a pullback attractor for such system as well. Both multivalued and single-valued cases are considered.Keywords: lattice dynamical systems, non-autonomous systems, di erential equations with delay, set-valued dynamical systems, pullback attractor MSC: 34K05, 34K31, 35B40, 35B41, 35K55, 35K40, 37L30, 58C06

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“…We also mention that, following Mao [14], many papers were devoted to improving the results of Mao [14] by weakening the Lipschitz conditions on coefficients (e.g., see Jiang and Wang [15], Fei [16], Caraballo et al [17], Wu et al [18], Jiang and Shen [19], Taniguchi [20], Fan and Jiang [21], Bao and Hou [22], Ren et al [23], Taniguchi [24], Wang and Huang [25], Lin [26], Xie [27], Fei [28], Ren and Zhang [29] and Zhang [30], etc.). In particular, Caraballo et al [17], Taniguchi [20], Bao and Hou [22], and Taniguchi [24] have studied the existence and uniqueness of solutions to SDEs under the non-Lipschitzian condition.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, Caraballo et al [17], Taniguchi [20], Bao and Hou [22], and Taniguchi [24] have studied the existence and uniqueness of solutions to SDEs under the non-Lipschitzian condition.…”

Section: Introductionmentioning

confidence: 99%

Existence and Uniqueness of Solution for a Class of Stochastic Differential Equations

Cao

Huang²,

Zeng

2013

The Scientific World Journal

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A class of stochastic differential equations given by dx(t) = f(x(t))dt + g(x(t))dW(t), x(t 0) = x 0, t 0 ≤ t ≤ T < +∞, are investigated. Upon making some suitable assumptions, the existence and uniqueness of solution for the equations are obtained. Moreover, the existence and uniqueness of solution for stochastic Lorenz system, which is illustrated by example, are in good agreement with the theoretical analysis.

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Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities

Cited by 112 publications

References 54 publications

Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

Attractors for non-autonomous retarded lattice dynamical systems

Existence and Uniqueness of Solution for a Class of Stochastic Differential Equations

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