Decay estimates for solutions of nonlocal semilinear equations

TL;DR

This paper derives sharp algebraic decay estimates for solutions of nonlocal semilinear equations, linking the decay rate to the singularity of the elliptic Fourier multiplier's symbol, with applications to internal solitary waves.

Contribution

It provides a precise relation between symbol singularity and solution decay, extending understanding of nonlocal equations like the Benjamin-Ono equation.

Findings

Established sharp algebraic decay rates in weighted Sobolev norms.

Linked decay rates to the singularity of the symbol at the origin.

Applied results to the Benjamin-Ono equation for internal solitary waves.

Abstract

We investigate the decay for $|x|\rightarrow \infty$ of weak Sobolev type solutions of semilinear nonlocal equations $Pu=F(u)$. We consider the case when $P=p(D)$ is an elliptic Fourier multiplier with polyhomogeneous symbol $p(\xi)$ and derive sharp algebraic decay estimates in terms of weighted Sobolev norms. In particular, we state a precise relation between the singularity of the symbol at the origin and the rate of decay of the corresponding solutions. Our basic example is the celebrated Benjamin-Ono equation {equation} \label{BO}(|D|+c)u=u^2, \qquad c>0,{equation} for internal solitary waves of deep stratified fluids. Their profile presents algebraic decay, in strong contrast with the exponential decay for KdV shallow water waves.