WELL-POSEDNESS OF THE MODIFIED BENJAMIN-BONA-MAHONY EQUATION

Abstract

In this document, we have thoroughly discussed the rigorous mathematical aspects of the well-posedness of PDEs taking the modified Benjamin–Bona–Mahony (MBBM) equation as the reference. Here we focus on the conditions that enable the equation to meet the three criteria of well-posedness: first to exist, then to be unique, and finally to be stable (that is, continuous dependence on initial data). R and T represent, respectively, the real line and periodic domain in the analysis; the explicit mention of the distinction indicates the way that domain constraints affect well-posedness. Our investigation is carried out in Sobolev spaces and their embedding into Lebesgue spaces, covering all possible nonlinearities ∂x(up for p = 1,2,3. The work has shown that the intensity of nonlinearity and the shape of the domain impact the regularity requirements for well-posedness in each case.