  This paper focuses on the two-dimensional Benjamin-Bona-Mahony and
Benjamin-Bona-Mahony-Burgers equations with a general flux function. The aim is
at the global (in time) well-posedness of the initial-and boundary-value
problem for these equations defined in the upper half-plane. Under suitable
growth conditions on the flux function, we are able to establish the global
well-posedness in a Sobolev class. When the initial- and boundary-data become
more regular, the corresponding solutions are shown to be classical. In
addition, the continuous dependence on the data is also obtained.
