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Inertial manifolds for parabolic differential equations: The fully nonautonomous case

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    *Corresponding author 

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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  • 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.

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


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