Carleman estimates for third order operators of KdV and non KdV-type and applications

Abstract

In this paper we study a class of variable coefficient third order partial differential operators on ${\mathbb{R}}^{n+1}$, containing, as a subclass, some variable coefficient operators of KdV-type in any space dimension. For such a class, as well as for the adjoint class, we obtain a Carleman estimate and the local solvability at any point of ${\mathbb{R}}^{n+1}$. A discussion of possible applications in the context of dispersive equations is provided.