On the decay of solutions for the negative fractional KdV equation

TL;DR

This work analyzes the decay properties of solutions to the negative-dispersion fractional KdV equation $\partial_t u-\partial_x D^{a+1}u+u\partial_x u=0$ with $a\in(-\tfrac{5}{2},-2)$, focusing on propagation of polynomial weights $|x|^{\theta}$. By developing sharp weighted/derivative bounds for the linear and regularized groups and employing parabolic regularization, the authors establish local well-posedness and weight-persistence results in tailored weighted Sobolev spaces, showing that propagation can occur up to the threshold $\theta=\tfrac{1+2a}{2(1+a)}$, exceeding the positive-dispersion bound in some regimes. They also prove unique continuation principles, linking two-time decay in weighted spaces to precise conditions on the initial data, and they demonstrate sharp limits on decay via two-time arguments that force the initial data to vanish at the critical threshold. Overall, the paper reveals how negative dispersion alters the balance between nonlinearity and dispersion, yielding a complete framework for spatial decay in polynomial spaces for $-\tfrac{5}{2}<a<-2$.

Abstract

We explore the limits of fractional dispersive effects and their incidence in the propagation of polynomial weights. More precisely, we consider the fractional KdV equation when a differential operator of negative order determines the dispersion. We investigate what magnitude of weights and conditions on the initial data that allow solutions of the equation to persist in weighted spaces. As a consequence of our results, it follows that even in the presence of negative dispersion, it is still possible to propagate weights whose maximum magnitude is related to the dispersion of the equation. We also observe that our results in weighted spaces do not follow specific properties and limits that their counterparts with positive dispersion.

Theorems & Definitions (39)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Theorem 1.4
• Proposition 1.5
• Proposition 1.6
• Remark 1.7
• Lemma 2.1
• Proposition 2.2
• proof