Well-posedness and stability results for the Korteweg-de Vries-Burgers and Kuramoto-Sivashinsky equations with infinite memory: A history approach

The main concern of the present paper is to study the well-posedness and stabilityproblem of two different dispersive systems subject to the effect of a distributedinfinite memory term. The two systems are respectively governed by the one-dimensional Korteweg-de Vries-Burgers and Kuramoto-Sivashinsky equations ina bounded domain[0,1]. In order to deal with the presence of the memory term,we adopt the history approach. First, we show that both problems are well-posedin appropriate functional spaces by means of the Fixed-Point Theorem providedthat the initial condition is sufficiently small. Then, the energy method enables usto provide a decay estimate of the systems' energy according to the assumptionssatisfied by the physical parameters and the memory kernel.(c) 2022 Elsevier Ltd. All rights reserved