On the weak solutions and persistence properties for the variable depth KDV general equations

Considered in this paper is a nonlinear evolution equation for surface waves in shallow water over uneven bottom. Due to the dependence on the bottom function of the coefficients and its involved complicated structure of the equation, the generalized Gronwall's inequalities are proposed to be used to derive the required priori estimates, with which, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs(R) with 1