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2024 Evolution Families of $(X,Y,\varphi)$-type and Periodic Solutions to Nonautonomous Evolution Equations
Ngoc Huy Nguyen, Thieu Huy Nguyen, Thi Ngoc Ha Vu
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Taiwanese J. Math. Advance Publication 1-22 (2024). DOI: 10.11650/tjm/240607


Consider nonautonomous evolution equation $\dot{u} = A(t) u(t) + Bg(u)(t)$ in which the family of operators $(A(t))_{t \geq 0}$ generates the evolution family $(\mathcal{U}(t,s))_{t \geq s \geq 0}$ of $(X,Y,\varphi)$-type, i.e., $\|\mathcal{U}(t,0)x\|_{Y} \leq \varphi(t) \|x\|_{X}$, $t > 0$, for certain couple of Banach spaces $(X,Y)$ and real-valued, positive function $\varphi$ satisfying $\lim_{t \to \infty} \varphi(t) = 0$. Inspired by Serrin's technique, we develop a unified approach toward the problems on the existence of periodic solutions to above equation. As illustrations of our abstract results, we give applications to the existence and uniqueness of periodic solutions to Oseen–Navier–Stokes and damped wave equations, as well as the existence of local stable manifolds nearby the periodic solution to the damped wave equations.

Funding Statement

This work is financially supported by the Vietnamese National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2021.04.


We would like to thank the reviewer for careful reading of our manuscript. His or her comments, remarks, corrections and suggestions lead to improvement of the paper.


Download Citation

Ngoc Huy Nguyen. Thieu Huy Nguyen. Thi Ngoc Ha Vu. "Evolution Families of $(X,Y,\varphi)$-type and Periodic Solutions to Nonautonomous Evolution Equations." Taiwanese J. Math. Advance Publication 1 - 22, 2024.


Published: 2024
First available in Project Euclid: 26 June 2024

Digital Object Identifier: 10.11650/tjm/240607

Primary: 35B10 , 35Q30 , 76D07

Keywords: Damped wave equations , evolution families of $(X,Y,\varphi)$-type , local stable manifolds , Oseen–Navier–Stokes equations , periodic solutions , rotating and translating obstacle

Rights: Copyright © 2024 The Mathematical Society of the Republic of China

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