Initial-boundary value problems for nonlinear dispersive equations of evolution of order 2l+1, l is an element of N with a convective term of the form u(k)u(x), k is an element of N have been considered on intervals (0, L), L is an element of(0, +infinity). Definitions of regular and critical values of k as functions of l have been given. The existence and uniqueness of global regular solutions as well as exponential decay of them for small initial data have been established.