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Dichotomy Gap Conditions, Admissible Spaces, and Inertial Manifolds

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Abstract

We study the existence of an inertial manifold for the fully non-autonomous evolution equation of the form

$$\begin{aligned} \frac{du}{dt} + A(t)u(t) = f(t,u),\, t\in \mathbb {R}, \end{aligned}$$

in certain admissible spaces. We prove the existence of such an inertial manifold in the cases 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 \(\varphi \)-Lipschitz condition, i.e., \(\left\| f(t,x_1)-f(t,x_2)\right\| \leqslant \varphi (t)\left\| A(t)^{\theta } (x_1-x_2)\right\| \), where \(\varphi (\cdot )\) belongs to some admissible function space such that certain dichotomy gap condition holds. This dichotomy gap condition, on the one hand, extends the spectral gap condition known in the case of autonomous equations, on the other hand, provides a chance to come over the restricted spectral gap condition.

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Acknowledgements

This work is financially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.02-2021.04.

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  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

    Thieu Huy Nguyen & Thi Ngoc Ha Vu

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  1. Thieu Huy Nguyen
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  2. Thi Ngoc Ha Vu
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Correspondence to Thieu Huy Nguyen.

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Nguyen, T.H., Vu, T.N.H. Dichotomy Gap Conditions, Admissible Spaces, and Inertial Manifolds. J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-023-10320-z

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