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Well-Posedness of Mild Solutions to Stochastic Parabolic Partial Functional Differential Equations

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Abstract

In this paper, we study the well-posedness of mild solutions to stochastic parabolic partial functional differential equations with space–time white noise. Firstly, we establish an existence–uniqueness theorem under the global Lipschitz condition and the linear growth condition. Secondly, we show the existence–uniqueness property under the global/local Lipschitz condition but without assuming the linear growth condition. In particular, we consider the existence and uniqueness under the weaker condition than the Lipschitz condition. Finally, we obtain the nonnegativity and comparison theorems and utilize them to investigate the existence of nonnegative mild solutions under the linear growth condition without assuming the Lipschitz condition.

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Correspondence to Chaoliang Luo.

Additional information

Communicated by Ahmad Izani Md. Ismail.

This research was partially supported by grants from the National Natural Science Foundation of China (No. 11671123), Natural Science Foundation of Hunan Province (No. 14JJ1025), and Science Research Project of Hunan Province Education Office (Nos. 17C0467 and 17C0468).

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Luo, C., Guo, S. & Hou, A. Well-Posedness of Mild Solutions to Stochastic Parabolic Partial Functional Differential Equations. Bull. Malays. Math. Sci. Soc. 42, 355–379 (2019). https://doi.org/10.1007/s40840-018-0639-4

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  • DOI: https://doi.org/10.1007/s40840-018-0639-4

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