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 H01 with exponential growth of the external force.