Local well-posedness and persistence property for the generalized Novikov equation (Q379755)

The authors study the generalized Novikov equation NEWLINE\[NEWLINEu_t-u_{xxt}+(k+2) u^k u_x=(k+1) u^{k-1} u_x u_{xx} +u^k u_{xxx},NEWLINE\]NEWLINE for \(x\in \mathbb R, t>0\), \(u(x,0)=u_0(x)\), \(x\in\mathbb R\), which describes the motion of shallow water waves. Using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, they prove that the Cauchy problem for the generalized Novikov equation is locally well-posed in the Besov space \(B_{p,r}^s\) with \(1\leq p\), \(r\leq +\infty\) and \(s>\max (1+1/p,3/2)\). They also show the persistence property of the strong solutions which implies that the solution decays at infinity in the space variable provided that the initial function does.