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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 $$\cases dx(t)=[Ax(t)+f(t,x(t-\tau(t)))]\,dt+g(t,x(t-\delta(t)))\,dW(t),\quad & t\ge 0,\\ x_0(\cdot)=\xi\in D^b_{{\cal F}_0}([m(0),0],H)\endcases$$ 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 93E03 General theory of stochastic systems 60H10 Stochastic ordinary differential equations
Full Text:
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