Comportamiento asintótico en variable espacial de la solución de una perturbación no local de la ecuación de Benjamin-Ono.

Abstract

In this work, we study the initial value problem of a non-local perturbation of the Benjamin-Ono equation, which is expressed in the following form: ∂_t u (t, x) + H ∂_x ^ 2 u (t, x) ∂_x u (t, x) + ηH (∂_x u (t, x) + ∂_x ^ 3 u (t, x) ) = 0, (t, x) ∈ (0, T] × ℝ, u (0, x) = u_0 (x), x∈ℝ, where 0 <T ≤ + ∞. In general terms, we study the existence of solutions to the problem, the uniqueness and regularity of mild solutions in the frame of the non-negative order non-homogeneous Sobolev spaces and the most important contribution of this work lies in the study of the persistence of pointwise decay in spatial variable of the mild solutions. The existence and uniqueness proof is based on Picard's fixed point theorem and the regularity study takes advantage of the structure of the mild solutions. The analysis of persistence properties follows ideas previously seen in the article by F. Cortez and O. Jarrín: On decay properties and asymptotic behavior of solutions to a non-local perturbed KdV equation.