On decay properties and asymptotic behavior of solutions to a non-local perturbed KdV equation. (October 2019) - Cortez, Manuel Fernando; Jarrín, Oscar
Abstract: We consider the KdV equation with an additional non-local perturbation term defined through the Hilbert transform, also known as the OST-equation. We prove that the solutions u ( t, x ) have a pointwise decay in spatial variable: | u ( t, x ) | ≲ 1 1 + | x | 2, provided that the initial data has the same decaying and moreover we find the asymptotic profile of u ( t, x ) when | x | → + ∞ . Next, we show that decay rate given above is optimal when the initial data is not a zero-mean function and in this case we derive an estimate from below 1 | x | 2 ≲ | u ( t, x ) | for | x | large enough. In the case when the initial datum is a zero-mean function, we prove that the decay rate above is improved to 1 1 + | x | 2 + ε for 0 < ε ≤ 1 . Finally, we study the local-well posedness of the OST-equation in the framework of Lebesgue spaces.