Dynamics of stochastic nonlocal partial differential equations

Abstract

This paper is concerned with the asymptotic behavior of solutions to nonlocal stochastic partial differential equations with multiplicative and additive noise driven by a standard Brownian motion, respectively. First of all, the stochastic nonlocal differential equations are transformed into their associated conjugated random differential equations, we then construct the dynamical systems to the original problems via the properties of conjugation. Next, in the case of multiplicative noise, we establish the existence of the random attractor when it absorbs every bounded deterministic set. Particularly, it is shown the pullback random attractor, which is also forward attracting, becomes a singleton when the external forcing term vanishes at zero. Eventually, in the case of additive noise, two approaches are applied to prove the existence of pullback random attractors with the help of energy estimations. Actually, these two attractors turn out to be the same one.

Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and Wong--Zakai Approximations

“…with initial value v δ (τ ) = u δ (τ ) − φy δ (θ τ ω) := v 0,δ . In a similar way as [33,Theorem 7], we are able to prove that, problem (5.11) with initial value v 0,δ ∈ H and Dirichlet boundary condition possesses a unique weak solution,…”

“…However, as it is well known, the theory of random dynamical systems has only been applied successfully to problems modeled by partial differential equations when the noise possesses a particular form: additive or multiplicative noise. These two cases have already been analyzed in [33]. Recently, B. X. Wang and his collaborators (see [17,15,22]) have been using an idea to approximate the nonlinear noise by a stochastic process (called colored noise), which basically is a Wong-Zakai approximation of the derivative of the Wiener process, providing a rigorous approximation of the cases with additive and multiplicative noise (as we explained in the Introduction).…”

“…As we mentioned before, since it is not known how to apply the theory of random dynamical systems to study the long time behavior of problem (1.1), we have applied an approximation of this problem in Section 4 by using colored noise and proved that the approximate problem possesses a random attractor. In the next two sections, we will consider two particular cases of equation (1.1) which have been analyzed already within the framework of random dynamical systems (see [33]). When the stochastic forcing term g(t, u(t))…”

“…Notice that, for the particular case in which the noise term is linear (additive or multiplicative), the existence of random attractors of (1.1) has been analyzed in [33] by exploiting the tools of the theory of random dynamical systems. However, when the noise is nonlinear, this theory cannot be applied in a suitable way because it is not proved yet that the stochastic problem (1.1) generates a random dynamical system.…”

Dynamical Systems of Differential Equations Based on Information Technology: Effects of Integral Step Size on Bifurcation and Chaos Control of Discrete Hindmarsh–Rose Models

“…If the value of the system parameters (branching parameters) changes smoothly, the system behavior suddenly changes "qualitatively" or topologically. ere are two main types of bifurcations: local bifurcations and global bifurcations [15]. Each of the two main types contains several subtypes, as shown in Figure 8.…”