The Existence of WeakD-Pullback Exponential Attractor for Nonautonomous Dynamical System

First, for a processU(t,τ)∣t≥τ, we introduce a new concept, called the weakD-pullback exponential attractor, which is a family of setsM(t)∣t≤T, for anyT∈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 existk,l>0such thatdist(U(t,τ)B(τ),M(t))≤ke-(t-τ); that is,M(t)pullback exponential attractsB(τ). Then we give a method to obtain the existence of weakD-pullback exponential attractors for a process. As an application, we obtain the existence of weakD-pullback exponential attractor for reaction diffusion equation inH01with exponential growth of the external force.