A Characterization of Backward Bounded Solutions

Authors

Minkyu Kwak Department of Mathematics, Chonnam National University, Gwangju, Republic of Korea https://orcid.org/0009-0009-6440-7567
Jihoon Lee Department of Mathematics, Chonnam National University, Gwangju, Republic of Korea https://orcid.org/0000-0003-3478-8572
Bataa Lkhagvasuren Department of Mathematics, Chonnam National University, Gwangju, Republic of Korea https://orcid.org/0000-0003-0285-1151

DOI:

https://doi.org/10.37256/cm.6420256174

Keywords:

inertial manifold, invariant attracting set, asymptotic behaviour of solution, infinite dimensional dynamical system

Abstract

We prove that the collection M_−∞ of backward bounded solutions for a semilinear evolution equation is the graph of an upper hemicontinuous set-valued function from the low Fourier modes to the higher Fourier modes, which is invariant and contains the global attractor. We also show that there exists a limit M_∞ of finite dimensional Lipschitz manifolds M_tgenerated by the time t-maps (t > 0) from the flat manifold M₀ with the Hausdorff distance and we find M_∞⊂ M_−∞. No spectral gap conditions are assumed.

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Published

2025-07-15

Issue

Contemporary Mathematics, Volume 6 Issue 4 (2025), 3846-5367

Section

Research Article

License

This work is licensed under a Creative Commons Attribution 4.0 International License.