A Characterization of Backward Bounded Solutions (Contemporary Mathematics, 2025)
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 Mt generated 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.
Keywords: inertial manifold; invariant attracting set; asymptotic behaviour of solution; infinite dimensional dynamical system