We consider differential systems with memory terms, by convolution integrals, which account for the past history of one or more variables. The aim of this work is to analyze the passage to the singular limit when the memory kernel collapses into a Dirac mass. In particular, we focus on the reactiondiffusion equation with memory, and we discuss the convergence of solutions on finite time-intervals. When enough dissipativity is present, we also establish convergence results of the global and exponential attractors. Nonetheless, the techniques here devised are quite general, and suitable to be applied to a large variety of models.
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Conti et al. (2006) studied this question.