On decay properties and asymptotic behavior of solutions to a non-local perturbed KdV equation

Abstract

We consider the \emph{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 $u(t,x)$ has a pointwise decay in spatial variable: $\vert u(t,x)\vert \lesssim \frac{1}{1 + |x|^{2}}$, provided that the initial data has the same decaying and moreover we find the asymptotic profile of $u(t,x)$ when $|x| \to +\infty$. 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 $\frac{1}{\vert x\vert^2} \lesssim \vert u(t,x)\vert$ for $\vert x \vert$ large enough. In the case when the initial datum is a zero-mean function, we prove that the decay rate above is improved to $\frac{1}{1+\vert x \vert^{2+\varepsilon}}$ for $0<\varepsilon \leq 1$. Finally, we study the local-well posedness of the OST-equation in the framework of Lebesgue spaces.