Global well-posedness and stability of the mild solutions for a class of stochastic partial functional differential equations

The aim of this paper is to develop some theories of stochastic partial functional differential equations (SPFDEs) driven by infinite dimensional Wiener processes under the quasi-local Lipschitz condition and the restriction growth condition. Firstly, we establish the existence-uniqueness theorem of the global mild solutions for SPFDEs by using the intercept technique. Then, we discuss asymptotic behavior of the solutions. Furthermore, some criteria of exponential stability in the mean square are obtained by using Lyapunov method. An example is provided to show the effectiveness of the theoretical results.