Weighted identities for solutions of generalized Korteweg-de Vries equations

Consider the Korteweg-de Vries equation u _t + u _xxx + uu _x = 0 and its generalization u _t + u _xxx + f(u)x = 0. For the solutions of these equations, weighted identities (differential and integral) are obtained. These identities make it possible to establish the blow-up (in finite time) of the solutions of certain boundary-value problems.