Lower bounds for non-trivial travelling wave solutions of equations of KdV type (Q2888605)

The authors prove that every real \(C^1\) solution of a Korteweg-de Vries (KdV) equation which is bounded from above by a travelling wave whose amplitude decays fast enough is identically equal to 0. The authors consider the KdV equation \((\partial _{t}+\partial _{x}^3)u=a(u)\partial _{x}u\) where \(a\) is a smooth real function satisfying \(| a(s)| \leq M_1(| s| +| s| ^{j})\), \(j=1,2,3,\dots \). The solution \(u\) is supposed to satisfy \( \sup_{t\geq 0}\| u(.,t)\| _{H^1}^2\leq M_2\), \( \sup_{t\geq 0}\int e^{| x-bt| }| u(x,t)| ^2dx\leq M_3\) and \(\sup_{t\geq 0}\int e^{2\lambda | x-bt| }| u(x,t)| ^2dx<+\infty \), for some \(\lambda \) large enough.NEWLINENEWLINEFor the proof of this result, the authors introduce the function \(f=e^{\lambda \theta }u\) where \(\theta (x,t)=\varphi (| x-bt| )\) for some function \(\varphi \) which is deduced from a Carleman weight. A key tool for the proof is Kato's theory for KdV equations. The main steps of the proof consist in computing lower bounds for commutators for \(\lambda \) large enough. The authors here follow a previous study by the same authors, plus \textit{L. Escauriaza} [Commun. Math. Phys. 305, No. 2, 487--512 (2011; Zbl 1219.35203)], but the techniques which are used here are slightly different from the ones previously used. The authors give examples of KdV equations which satisfy the imposed hypotheses.