Chapter III - Global spectral asymptotics

Summary

Global spectral asymptotics of trajectories

We shall construct for all t ≥ 0 asymptotics of trajectories of a semigroup {S_t} which is uniform in respect to the bounded set B of initial data u₀ = u(0). We suppose that the semigroup {S_t}, S_t : E → E, has a finite number of equilibrium points. (Some of them might be unstable.) Any trajectory of such a semigroup tends as t → +∞ to one of the equilibrium points and has in the neighbourhood of this point a local spectral asymptotic, which was studied in §3. An asymptotic for a trajectory u(t), uniform with respect to initial functions u₀ = u(0), can be different from the individual one studied in §3 because, while u(t) → z^m if t → +∞, this trajectory u(t) can, for arbitrarily long times (different for different u₀ ∈ B), stay in neighbourhoods of unstable equilibrium points zⁱ ≠ z^m. So, for uniform approximation of u(t), we must use trajectories not only lying on M₊(z^m, ρ_m) but also lying on other manifolds M₊(zⁱ, ρ_i), i ≠ m. Therefore the constructed approximation curve has discontinuities (points of discontinuity lie in neighbourhoods of unstable equilibrium points).

Here we describe only a general schema of construction of global spectral asymptotics. For detailed description of these results, see Babin & Vishik [4], [1].

Definition 5.1.

Let_δ() be a δ-neighbourhood of the set of equilibrium points of {S_t} <I>and let B ⊆ E be a bounded set.