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)|≲11+|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 11+|x|2+ε for 0<ε≤1. Finally, we study the local-well posedness of the OST-equation in the framework of Lebesgue spaces.
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