On the Well-Posedness of a Class of Whitham-like Nonlocal Equations with Weak Dispersion

Abstract

We consider the Cauchy problems for a class of Whitham-like nonlocal equations with weak dispersion. Specifically, based on classical theory by Kato, local well-posedness in Sobolev spaces of order s>3/2 for this class of equations is proven, both on the real line and on the torus. The possibility of extending to global well-posedness is also discussed, and in one specific case a global ill-posedness result is given. Additionally, the text includes a largely self-contained treatment of the theory of Sobolev spaces of real order, both on R^d and on the one-dimensional torus