Inertial manifolds for parabolic differential equations: The fully nonautonomous case

Nguyen Thieu Huy; Pham Truong Xuan; Vu Thi Ngoc Ha; Vu Thi Thuy Ha

doi:10.3934/cpaa.2022005

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2022, Volume 21, Issue 3: 943-958. Doi: 10.3934/cpaa.2022005

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Inertial manifolds for parabolic differential equations: The fully nonautonomous case

1.
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi, 1 Dai Co Viet, Hanoi, Vietnam
2.
Faculty of Computer Science and Engineering, Department of Mathematics, Thuyloi University, Khoa Cong nghe Thong tin, Bo mon Toan, Dai hoc Thuy loi, 175 Tay Son, Dong Da, Ha Noi, Vietnam
3.
Fermat Education, 6A1-Ngoc Khanh, Ba Dinh, Hanoi, Vietnam

^*Corresponding author
^*Corresponding author

Received: June 30, 2021

Revised: October 31, 2021

Early access: December 2021

Published: March 2022

The work of V.T.N. Ha is partly supported by by the Project of Vietnam Ministry of Education and Training under Project B2022-BKA-06. This work is financially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.02-2021.04

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Abstract

We study the existence of an inertial manifold for the solutions to fully non-autonomous parabolic differential equation of the form

$ \dfrac{du}{dt} + A(t)u(t) = f(t,u),\, t> s. $

We prove the existence of such an inertial manifold in the case that the family of linear partial differential operators $ (A(t))_{t\in { \mathbb {R}}} $ generates an evolution family $ (U(t,s))_{t\ge s} $ satisfying certain dichotomy estimates, and the nonlinear forcing term $ f(t,x) $ satisfies the Lipschitz condition such that certain dichotomy gap condition holds.

Keywords:

Fully non-autonomous parabolic differential equations,
inertial manifolds,
dichotomy gap condition,
Lyapunov-Peron methods.

Mathematics Subject Classification: Primary: 35B42; 35B35.

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