Well-posedness and continuity properties of the Fornberg–Whitham equation in Besov spaces

Abstract

In this paper, we prove well-posedness of the Fornberg–Whitham equation in Besov spaces B2,rs in both the periodic and non-periodic cases. This will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data-to-solution map provided that the initial data belongs to B2,rs. We also establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from any bounded subset of B2,rs to C([−T,T];B2,rs). Furthermore, we prove a Cauchy–Kowalevski type theorem for this equation that establishes the existence and uniqueness of real analytic solutions and also provide blow-up criterion for solutions.