The Existence of Weak D-Pullback Exponential Attractor for Nonautonomous Dynamical System.

ABSTRACT

First, for a process {U(t, τ)∣t ≥ τ}, we introduce a new concept, called the weak D-pullback exponential attractor, which is a family of sets {ℳ(t)∣t ≤ T}, for any T ∈ ℝ, satisfying the following (i) ℳ(t) is compact, (ii) ℳ(t) is positively invariant, that is, U(t, τ)ℳ(τ) ⊂ ℳ(t), and (iii) there exist k, l > 0 such that dist(U(t, τ)B(τ), ℳ(t)) ≤ ke (-(t-τ)); that is, ℳ(t) pullback exponential attracts B(τ). Then we give a method to obtain the existence of weak D-pullback exponential attractors for a process. As an application, we obtain the existence of weak D-pullback exponential attractor for reaction diffusion equation in H 0 (1) with exponential growth of the external force.