Abstract
Abstract. In this paper, we investigate the time of existence of the solutions to two full dispersion models derived from the water waves equations in the shallow water regime: the Whitham equation and a Whitham–Boussinesq system in dimensions one and two. The regime is characterized by the nonlinearity parameter [Formula: see text] and the shallow water parameter [Formula: see text]. We extend the lifespan of the solution beyond the hyperbolic time [Formula: see text]. More precisely, we establish well-posedness on the timescale of order [Formula: see text] in the one-dimensional case and of order [Formula: see text] in dimension two. We emphasize that for the two-dimensional case, we obtain a time of existence of order [Formula: see text] in the long wave regime [Formula: see text]. This kind of result seems to be new, even for the Boussinesq systems. The proofs combine energy methods with Strichartz estimates. Here, a key ingredient is to obtain new refined Strichartz estimates that include the small parameter [Formula: see text]. These techniques are robust and could be adapted to improve the lifespan of solutions for other equations and systems of the same form.