Narrow Results

This paper studies the growth and decay rates of solutions of scalar stochastic delay differential equations of Itô type. The equations studied have a linear drift which contains an unbounded delay term, and a nonlinear diffusion term, which depends on the current state only. We show that when the nonlinearity at zero or infinity is sufficiently weak, the same non-exponential decay and growth rates found in the corresponding underlying linear deterministic equation are recovered, in an almost sure sense.

The theory of pullback attractors for multi-valued non-compact random dynamical systems and a method of asymptotic compactness based on the concepts of the Kuratowski measure of the non-compactness of a bounded set are used to prove the existence of pullback attractors for the multi-valued non-compact random dynamical systems associated with the semi-linear degenerate parabolic unbounded delay equations with both deterministic and random external terms.

We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order $p\ge 2$ driven by a tempered fractional Brownian motion (TFBM) $B^{\sigma ,\lambda }(t)$ with $-1/2<\sigma <0$ and $\lambda >0$. 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 $\lambda \to 0$, $\sigma \in (-1/2,0)$ or $\lambda >0$, $\sigma \to {\sigma }_0\in (-1/2,0)$. 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.

In this paper, the global exponential stability of fuzzy cellular neural networks (FCNNs) with variable coefficients and unbounded delays was investigated. Without assuming the boundedness and differentiability of the activation functions, based on the properties of M-matrix, by constructing vector Lyapunov functions and applying differential inequalities, the sufficient conditions ensuring globally exponential stability of the equilibrium point of fuzzy cellular neural networks with variable coefficients and unbounded delays are obtained.