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

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 𝐻1 0 with exponential growth of the external force.