On local energy decay for large solutions of the Zakharov-Kuznetsov\n equation

2020/07/09 by Argenis J. Méndez, Méndez, Argenis, Claudio Muñoz +5 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Nonlinear Photonic Systems

paper · pdf · doi:10.48550/arxiv.2007.04918

Abstract

We consider the Zakharov-Kutznesov (ZK) equation posed in mathbb R^d, with\nd=2 and 3. Both equations are globally well-posed in L²( mathbb R^d). In\nthis paper, we prove local energy decay of global solutions: if u(t) is a\nsolution to ZK with data in L²( mathbb R^d), then \
liminf_{t
rightarrow\n
infty}
int
_Omegad(t)u²(
bf x,t)
mathrmd
bf x=0, for suitable\nregions of space \Ω_d(t)\⊆ mathbb R^d around the origin, growing\nunbounded in time, not containing the soliton region. We also prove local decay\nfor H¹( mathbb R^d) solutions. As a byproduct, our results extend decay\nproperties for KdV and quartic KdV equations proved by Gustavo Ponce and the\nsecond author. Sequential rates of decay and other strong decay results are\nalso provided as well.\n

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