Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications

Qi Yao; Linshan Wang; Yangfan Wang

doi:10.3934/dcdsb.2020310

Article Contents

2021, Volume 26, Issue 9: 4727-4743. Doi: 10.3934/dcdsb.2020310

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Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications

1.
School of Mathematical Sciences, Ocean University of China, Qingdao, Shandong 266100, China
2.
Department of Mathematics, University of Dundee, Dundee DD1 4HN, UK
3.
Ministry of Education Key Laboratory of Marine Genetics and Breeding, College of Marine Life Science, Ocean University of China, Qingdao, Shandong 266100, China

^* Corresponding authors: Linshan Wang and Yangfan Wang
^* Corresponding authors: Linshan Wang and Yangfan Wang

Received: December 31, 2018

Revised: May 31, 2020

Early access: October 2020

Published: September 2021

The authors are supported by the China Scholarship Council under Grant 201906330009, the Fundamental Research Funds for the Central Universities under Grant 201861005, the National Key Research and Development Program of China under Grant 2018YFD0901601, the National Natural Science Foundation of China under Grant 11771014 and Grant 31772844, and the Major Basic Research Projects of Shandong Natural Science Foundation under Grant 2018A07

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Abstract

In this paper, we focus on the mild periodic solutions to a class of delayed stochastic reaction-diffusion differential equations. First, the key issues of Markov property in Banach space $ C $, $ p $-uniformly boundedness, and $ p $-point dissipativity of mild solutions $ \boldsymbol{u}_t $ to the equations are discussed. Then, the theorems of existence-uniqueness and exponential stability in the mean-square sense of the mild periodic solutions are established by using the dissipative theory and the operator semigroup technique, and the relevant results about the existence of mild periodic solutions in the quoted literature are generalized. Next, the given theoretical results are successfully applied to the delayed stochastic reaction-diffusion Hopfield neural networks, and some easy-to-test criteria of exponential stability for the mild periodic solution to the networks are obtained. Finally, some examples are presented to demonstrate the feasibility of our results.

Keywords:

Delayed stochastic reaction-diffusion differential equations,
mild periodic solutions,
existence-uniqueness,
stability,
neural networks.

Mathematics Subject Classification: Primary: 35B10, 34K20, 60H15; Secondary: 90B15.

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Figure 1. The periodic trajectory and simulation of $ u_1 $ and $ u_2 $ in Example 1

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Figure 2. The phase graph in Example 1

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Figure 3. The periodic trajectory and simulation of $ u_1 $ and $ u_2 $ in Example 1

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Figure 4. The trajectory of u to Example 3.1 (left) and Example 3.2 (right) in Example 3

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