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Existence and Stability for Stochastic Partial Differential Equations with Infinite Delay

DOI: 10.1155/2014/235937

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We consider a class of neutral stochastic partial differential equations with infinite delay in real separable Hilbert spaces. We derive the existence and uniqueness of mild solutions under some local Carathéodory-type conditions and also exponential stability in mean square of mild solutions as well as its sample paths. Some known results are generalized and improved. 1. Introduction The theory of stochastic partial differential equations (SPDEs) has recently become an important area of investigation stimulated by its numerous applications to problems arising in natural and social sciences. There is much current interest in studying qualitative properties for SPDEs (see, e.g., Caraballo et al. [1], Liu [2], Luo and Liu [3], Da Prato and Zabczyk [4], Peszat and Zabczyk [5], Wang and Zhang [6], and references therein). We would like to mention that the stochastic partial functional differential equations (SPFDEs) have been considered intensively. For example, under the global Lipschitz and linear growth conditions, Govindan [7] showed by the stochastic convolution the existence, uniqueness, and almost sure exponential stability of neutral SPDEs with finite delays; Taniguchi et al. [8] considered the existence and uniqueness of mild solutions to SPDEs with finite delays by Banach fixed point theorem; while by imposing a so-called Carathéodory condition on the nonlinearities, Jiang and Shen [9] studied the existence and uniqueness of mild solutions for neutral SPFDEs by successive approximation; Samoilenko et al. [10] investigated the existence, uniqueness, and controllability results for neutral SPFDEs. On the other hand, it is well known that infinite delay (stochastic) equations have wide application in many areas [11, 12]. However, as for neutral SPDEs with infinite delay, as far as we know, there exist only a few results about the existence and asymptotic behavior of mild solutions. We mention here the recent papers by Ren and Sun [13] and Li and Liu [14] considering the existence of solutions of second-order stochastic evolution equations and neutral stochastic differential inclusions with infinite delay, respectively; Cui and Yan [15] investigated the existence and longtime behavior of mild solutions for a class of neutral stochastic partial differential equations with infinite delay in distribution, while Taniguchi [16] concerned the existence and asymptotic behavior for stochastic evolution equations with infinite delay. In this paper, inspired by the aforementioned papers [13, 15], we consider a class of neutral stochastic partial differential

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