Remarks on equations of Benjamin–Bona–Mahony type

We investigate an initial-boundary value problem for equations of Benjamin–Bona–Mahony (BBM) type in two different physical situations. In the first, the mixed problem is considered on a cylinder domain Q of Rn×Rt. In the second one, the mixed problem is studied inside of an increasing noncylindrical domain of Rn×Rt. In both situations we show the existence of a unique nonlocal solution. In cylindrical case it is proved the existence of weak and strong solutions, regularity of strong solutions, and in noncylindrical case weak solutions. One of the goals of this paper is to show that the noncylindrical problem is well-posed by using the penalty method idealized by Lions [J.L. Lions, Une remarque sur les problèmes d'évolution non linéaires dans des domaines non cylindriques, Rev. Roumaine Math. Pures Appl. 9 (1964) 11–18].