On odd-order quasilinear evolution equations with general nonlinearity on a bounded interval

Initial-boundary value problem, posed on a bounded interval, is considered for a class of odd-order evolution equations with general nonlinearity. Assumptions on the equations do not provide global a priori estimate, for example, for the initial value problem in comparison with the Korteweg-de Vries equation. However, internal dissipativity of the equations implies results on global well-posedness and large-time decay of small solutions. Considered class includes the Kaup-Kuperschmidt and other important equations. Inverse problems with integral overdetermination are also considered.