Continuous families of exponential attractors for singularly perturbed equations with memory

We establish, for a family of semigroups S_(ε:) H_ε→H_ε depending on aperturbation parameter ε∈[0,1], where the perturbation is allowed tobecome singular at ε=0, a general theorem on the existenceof exponential attractors E_ε satisfying a suitableHölder continuity property with respect tothe symmetric Hausdorff distanceat every ε∈[0,1]. The result is appliedto the abstract evolution equations with memoryx_t (t)+∫_0^∞▒█(k_ε (s) B_0 (x(t-s) )ds+B_1 (x(t) )=0, ε∈(0,1] )where k_ε (s)=1/ε k(s/ε)is the rescaling of a convex summable kernel k with unit mass.Such a family can be viewed as a memory perturbationof the equation x_t (t)+B_0 (x(t) )+B_1 (x(t) )=0formally obtained in the singular limit ε→0.