# American Institute of Mathematical Sciences

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

Received  July 2021 Revised  November 2021 Early access December 2021

Fund Project: 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

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.
Citation: Nguyen Thieu Huy, Pham Truong Xuan, Vu Thi Ngoc Ha, Vu Thi Thuy Ha. Inertial manifolds for parabolic differential equations: The fully nonautonomous case. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2022005
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