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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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