DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 200–208 SINGULAR LIMIT OF DISSIPATIVE HYPERBOLIC EQUATIONS WITH MEMORY

Abstract. We consider a class of weakly damped semilinear hyperbolic equations with memory, expressed by a convolution integral. We study the passage to the singular limit when the memory kernel collapses into the Dirac mass at zero, and we establish a convergence result for a proper family of exponential attractors. 1. Formulation of the Problem. Let Ω ⊂ R3 be a bounded domain with smooth boundary ∂Ω. For u = u(x, t) : Ω × R → R, we consider, on the time-interval R+ = (0,∞), the differential equation utt + αut − [β + kε(0)]∆u− 0 k′ε(s)∆u(t − s)ds+ φ(u) = f, (1) arising, for instance, in the theory of viscoelasticity (cf. [4, 9]). Here, α> 0, β> 0, φ: R → R is a nonlinear term of (at most) cubic growth, f: Ω → R is the external force, whereas the memory kernel kε(s) = 1