A higher dispersion KdV equation on the line. (October 2020) - Figueira, Renata; Himonas, A. Alexandrou; Yan, Fangchi
Abstract: The Cauchy problem for a Korteweg–deVries equation with dispersion of order m = 2 j + 1, where j is a positive integer, (KdVm), is studied with data in Sobolev and analytic spaces. First, optimal bilinear estimates in Bourgain spaces are proved and using them well-posedness in Sobolev spaces H s, s > − j + 1 4, is established. Then, well-posedness in analytic Gevrey spaces G δ, s, δ > 0, is proved by using an analytic version of the bilinear estimates. This implies that the uniform radius of analyticity persist for some time. For the later times a lower bound for the radius of spacial analyticity is derived, which is given by δ ( t ) ≥ c t − α, with α = 4 3 + ε, for any ε > 0, when j = 1, and α = 1 when j ≥ 2 . Finally, it is shown that the regularity of the solution in the time variable is Gevrey of order m, and this is optimal.