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In this paper, we consider stochastic evolution equations with finite delay driven by fractional Brownian motion (fBm) with Hurst index $ H\in(0,1) $. First, the global existence and uniqueness of mild solutions are established by using a temporally weighted norm, the semigroup theory, the Banach fixed point theorem, and a new estimate of the stochastic integral with respect to fBm. Then after extending the integral inequality to the general case, we obtain the mean-square exponential stability of mild solutions. In particular, the convolution method and the Young convolution inequality are utilized to deal with the difficulty of the noise term caused by fBm with Hurst index $ H\in(0,1) $. Finally, the stochastic evolution equation with multiple time-varying delays and the reaction diffusion neural networks system with time-varying delay are considered, and the global existence, uniqueness and exponential stability of mild solutions are proved by using the abstract results. [ABSTRACT FROM AUTHOR]

This paper is concerned with stochastic delay evolution equations driven by tempered fractional Brownian motion (tfBm) $ B_Q^{\sigma, \lambda}(t) $ with time fractional operator of order $ \alpha\in (1/2+\sigma, 1) $, where $ \sigma\in (-1/2, 0) $ and $ \lambda>0 $. First, we establish the global existence and uniqueness of mild solutions by using the new established estimation of stochastic integrals with respect to tfBm. Moreover, based on the relations between the stochastic integrals with respect to tfBm and fBm, we show the continuity of mild solutions for stochastic delay evolution equations when tempered fractional noise is reduced to fractional noise. Finally, we analyze the stability with general decay rate (including exponential, polynomial and logarithmic stability) of mild solutions for stochastic delay evolution equations with tfBm and time tempered fractional operator. [ABSTRACT FROM AUTHOR]

The polynomial stability problem of stochastic delay differential equations has been studied in recent years. In contrast, there are relatively few works on stochastic partial differential equations with pantograph delay. The present paper is devoted to investigating large-time asymptotic properties of solutions for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. We first show that the mild solutions of stochastic pantograph delay evolution equations with nonlinear multiplicative noise tend to zero with general decay rate (including both polynomial and logarithmic rates) in the p th moment and almost sure senses. The analysis is based on the Banach fixed point theorem and various estimates involving the gamma function. Moreover, by using a generalized version of the factorization formula and exploiting an approximation technique and a convergence analysis, we construct the nontrivial equilibrium solution, defined for t ∈ ℝ , for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. In particular, the uniqueness, Hölder regularity in time and general stability, in the p th moment and almost sure senses, of the nontrivial equilibrium solution are established. [ABSTRACT FROM AUTHOR]

This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays and additive white noise. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor. [ABSTRACT FROM AUTHOR]

We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order p ≥ 2 driven by a tempered fractional Brownian motion (TFBM) B σ , λ (t) with − 1 / 2 < σ < 0 and λ > 0. First, the global existence and uniqueness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to TFBM in the sense of p th moment. Moreover, based on the relations between the stochastic integrals with respect to TFBM and fractional Brownian motion, we show the continuity of mild solutions in the case of λ → 0 , σ ∈ (− 1 / 2 , 0) or λ > 0 , σ → σ 0 ∈ (− 1 / 2 , 0). In particular, we obtain p th moment Hölder regularity in time and p th polynomial stability of mild solutions. This paper can be regarded as a first step to study the challenging model: stochastic 2D-Navier–Stokes equations with unbounded delay driven by tempered fractional Gaussian noise. [ABSTRACT FROM AUTHOR]

In this paper, fractionally dissipative 2D quasi-geostrophic equations with an external force containing infinite delay is considered in the space H s with s ≥ 2 - 2 α and α ∈ (1 2 , 1) . First, we investigate the existence and regularity of solutions by Galerkin approximation and the energy method. The continuity of solutions with respect to initial data and the uniqueness of solutions are also established. Then we prove the existence and uniqueness of a stationary solution by the Lax–Milgram theorem and the Schauder fixed point theorem. Using the classical Lyapunov method, the construction method of Lyapunov functionals and the Razumikhin–Lyapunov technique, we analyze the local stability of stationary solutions. Finally, the polynomial stability of stationary solutions is verified in a particular case of unbounded variable delay. [ABSTRACT FROM AUTHOR]

We consider the following lattice nonclassical diffusion equation with delays v̇ i(t)+λ0vi(t)+(-1)p Δpvi(t)+ε(-1)p Δp v̇ i(t)=fi(vi(t-ρ(t)))+gi(t), i∈ Z, where ε∈(0 1], λ0 is a positive constant with λ0<1, p is any positive integer and ▵ is the discrete one-dimensional Laplace operator. Under suitable conditions on f and g we prove the existence of pullback attractors for the multi-valued process associated with the ε-small perturbed systems for which the uniqueness of solutions need not hold. Moreover, we compare the dynamics of the original systems and the ε-small perturbed systems, and show that their attractors are "close" in the sense of Hausdorff semidistance. [ABSTRACT FROM AUTHOR]

The existence of a pullback attractor in a space of higher regularity is established for the non-autonomous non-compact dynamical system associated with the reaction-diffusion delay equation with both non-autonomous deterministic and stochastic forcing terms defined on all n-dimensional space. The asymptotical compactness of the solutions is demonstrated by using the uniform estimates for far-field values of solutions and a new method. [ABSTRACT FROM AUTHOR]