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 B 2 , r s 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 B 2 , r s . 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 B 2 , r s to C ( [ − T , T ] ; B 2 , r s ) . 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.