## Stability of stochastic partial differential equations with infinite delays.(English)Zbl 1151.60336

Summary: In this paper, we study the existence and the asymptotical stability in $$p$$-th moment of mild solutions to stochastic partial differential equations with infinite delays
$\begin{cases} dx(t)=[Ax(t)+f(t,x(t-\tau(t)))]\,dt+g(t,x(t-\delta(t)))\,dW(t),\quad & t\geq 0,\\ x_0(\cdot)=\xi\in D^b_{{\mathcal F}_0}([m(0),0],H)\end{cases}$
where $$t-\tau (t),t-\delta(t)\to\infty$$ with delays $$\tau (t),\delta(t)\to\infty$$ as $$t\to \infty$$. Our method for investigating the stability of solutions is based on the fixed point theorem.

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 93E03 Stochastic systems in control theory (general) 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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### References:

 [1] J.A.D. Appleby, Fixed points, stability and harmless stochastic perturbations, preprint [2] Burton, T.A., Stability by fixed point theory for functional differential equations, (2006), Dover Publications, Inc. New York · Zbl 1090.45002 [3] Caraballo, T., Asymptotic exponential stability of stochastic partial differential equations with delay, Stochastics, 33, 27-47, (1990) · Zbl 0723.60074 [4] Caraballo, T.; Liu, Kai, Exponential stability of mild solutions of stochastic partial differential equations with delays, Stochastic anal. appl., 17, 743-763, (1999) · Zbl 0943.60050 [5] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press · Zbl 0761.60052 [6] Govindan, T.E., Exponential stability in Mean-square of parabolic quasilinear stochastic delay evolution equations, Stochastic anal. appl., 17, 443-461, (1999) · Zbl 0940.60076 [7] Haussmann, U.G., Asymptotic stability of the linear Itô equation in infinite dimensions, J. math. anal. appl., 65, 219-235, (1978) · Zbl 0385.93051 [8] Ichikawa, A., Stability of semilinear stochastic evolution equations, J. math. anal. appl., 90, 12-44, (1982) · Zbl 0497.93055 [9] Liu, Kai, Lyapunov functionals and asymptotic stability of stochastic delay evolution equations, Stochastics, 63, 1-26, (1998) · Zbl 0947.93037 [10] Liu, Kai, () [11] Luo, Jiaowan, Fixed points and stability of neutral stochastic delay differential equations, J. math. anal. appl., (2007) · Zbl 1160.60020 [12] Mao, Xuerong, Exponential stability for stochastic differential delay equations in Hilbert spaces, Quart. J. math., 42, 77-85, (1991) · Zbl 0719.60062 [13] Taniguchi, T., Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics, 53, 41-52, (1995) · Zbl 0854.60051
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