2018/11/26 by Cortez, Manuel Fernando, Jarrín, Oscar · 1 citation
#35B40 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1811.10492
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) has a pointwise decay in spatial variable: \vert u(t,x)\vert \lesssim \frac11 + |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)/(\vert x\vert2) \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 \frac11+\vert x \vert2+ε for 0