Persistent Decay of Solutions to the k-abc Equation in Weighted $$L^p$$Lp Spaces

Abstract

In this paper, we study a k-abc equation with $$(k+1)$$-degree nonlinearities, which is a 4-parameter family of nonlocal evolution equations and contains some famous ones. By moderate weight functions from time-frequency analysis, we show some persistence results for solutions of the equation in weighted $$L^p_\phi := L^p({\mathbf {R}}, \phi ^p(x)dx)$$ spaces for a large class of moderate weights. To overcome the difficulty caused by the higher nonlinear term, we take full advantage of features of the admissible weight function $$\phi $$. Our results cover some works on persistence properties of the Camassa–Holm type equations.