An initial- and boundary-value problem for a model equation for propagation of long waves

An initial- and boundary-value problem for a model equation for small-amplitude long waves is shown to be well-posed. The model has the form u_t + u_x + uu_x ? vu_xx ? α²u_xxt = 0, where x? [0, 1] and t ? 0. The solution u = u(x, t) is specified at t = 0 and on the two boundaries x = 0 and x = 1. Unique classical solutions are shown to exist, which depend continuously on variations of the specified data within appropriate function classes.