The Cauchy problem for the dispersive Kuramoto-Velarde equation
initial value problem | Nonlinear Dispersive Equation
The purpose of this work is the study of the well-posedness of the initial value problem (IVP) associated to the dispersive Kuramoto-Velarde equation. In the dissipative case, we prove local well-posedness in Sobolev spaces $H^s(\R)$ for $s >-1$, and ill-posedness in $H^s(\R)$ for $s<-1$. In the purely dispersive case, we first prove an ill-posedness result, which states that the flow map data-solution cannot be of class $C^2$ in any Sobolev space $H^s(\R)$, for $s \in \R$. Then, we prove a well-posedness result in weighted Besov spaces for small initial data.