Nonlinear behavior of model equations which are linearly ill-posed

We discuss two model equations of nonlinear evolution which demonstrate that linearly ill-posed problems may be well-posed in a mild sense. For the nonlocal equation (1.4), smooth solutions exist for all time, are unique, and depend continuously on the initial data in low norms. For the partial differential equation (1.1), solutions always exist; we do not know whether they are unique, but if they are, they also have continuous dependence on data. The large-time behavior of solutions and other qualitative properties are discussed