Well-posedness and stability results for the Korteweg–de Vries–Burgers and Kuramoto–Sivashinsky equations with infinite memory: A history approach. (June 2022)

• Record Type: Journal Article
• Authors: Chentouf, Boumediène; Guesmia, Aissa
• Is Part Of: Nonlinear analysis. Volume 65(2022)
• Publication Date: 2022-06
• Subjects: Korteweg–de Vries–Burgers equation -- Kuramoto–Sivashinsky equation -- Infinite memory -- Well-posedness -- Stability -- Energy method; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2022.103508
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: The main concern of the present paper is to study the well-posedness and stability problem of two different dispersive systems subject to the effect of a distributed infinite memory term. The two systems are respectively governed by the one-dimensional Korteweg–de Vries–Burgers and Kuramoto–Sivashinsky equations in a 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-posed in appropriate functional spaces by means of the Fixed-Point Theorem provided that the initial condition is sufficiently small. Then, the energy method enables us to provide a decay estimate of the systems' energy according to the assumptions satisfied by the physical parameters and the memory kernel.