Dynamics of small solutions in KdV type equations: decay inside the linearly dominated region

Claudio Muñoz

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arXiv · 1802.05547

Dynamics of small solutions in KdV type equations: decay inside the linearly dominated region

Claudio Muñoz

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Abstract

In this paper we prove that all small, uniformly in time $L^1\cap H^1$ bounded solutions to KdV and related perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order $t^{1/2}$ around any compact set in space. This set is included in the linearly dominated dispersive region $x\ll t$. Moreover, we prove this result independently of the well-known supercritical character of KdV scattering. In particular, no standing breather-like nor solitary wave structures exists in this particular regime. For the proof, we make use of well-chosen weighted virial identities. The main new idea employed here with respect to previous results is the fact that the $L^1$ integral is subcritical with respect to the KdV scaling.

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Claudio Muñoz. 2018-02-15. Dynamics of small solutions in KdV type equations: decay inside the linearly dominated region. https://arxiv.org/abs/1802.05547

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