Weighted Stepanov-Like Pseudo Almost Automorphic Solutions for Evolution Equations and Applications
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Weighted Stepanov-Like Pseudo Almost Automorphic Solutions for Evolution Equations and Applications Nguyen Thieu Huy1 · Vu Thi Ngoc Ha1 · Le The Sac2 · Pham Truong Xuan2 Received: 30 July 2019 / Revised: 25 December 2019 / Accepted: 18 January 2020 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract In this paper, the existence and stability of (weighted) Stepanov-like pseudo almost automorphic solutions are proved for a large class of linear and semilinear evolution equations in interpolation spaces. Our method is based on the one hand on a combination of differential inequalities and interpolations functors for the case of linearized equation, and on the other hand on the fixed point argument for the semi-linear equations to handle the case of (weighted) Stepanov-like pseudo almost automorphic functions. Finally, we apply the abstract results to various problems of incompressible viscous fluid flows. Keywords Linear and semi-linear evolution equations · Stepanov-like and weighted Stepanov-like almost automorphic solutions Mathematics Subject Classification 2010 35B15 · 35B35 · 35Q30 · 76D05
1 Introduction The study of the periodic and almost periodic solutions and their generalizations to evolution equations is an important research direction related to long-time properties of solutions to Nguyen Thieu Huy
[email protected] Vu Thi Ngoc Ha [email protected] Le The Sac [email protected] Pham Truong Xuan [email protected] 1
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam
2
Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam
N.T. Huy et al.
evolution equations. For the case of periodic solutions, there are some standard approaches used for that research, such as Massera principle [37, 60], Tikhonovs fixed-point theorem [47], or the Lyapunov functionals [58] applied to some specific class of differential equations, and the most popular approaches for proving the existence of a periodic solution are the ultimate boundedness of solutions and the compactness of Poincar´e map realized through some compact embeddings (see [34, 47–49, 58] and the references therein). Furthermore, in the case of partial differential equations in unbounded domains or equations having unbounded solutions, such compact embeddings may not hold true any longer, and the existence of bounded solutions is difficult to obtain since the appropriate initial vector (or conditions) guaranteeing the boundedness of the solution starting from that vector is not easy to find. One method to overcome such difficulties is to use the so-called Masseratype theorem, roughly speaking, that if a differential equation has a bounded solution then it has a periodic one. Actually, this Massera’s methodology combined with interpolation spaces has been used to prove the existence of periodic solutions to equations arising from fluid dynamics in [30] and [22]. In those works, the interpolatio
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