Decay estimates for solutions of nonlocal semilinear equations

Abstract

We investigate the decay for $\left|x\right|\to \infty$ of weak Sobolev-type solutions of semilinear nonlocal equations $Pu=F\left(u\right)$. We consider the case when $P=p\left(D\right)$ is an elliptic Fourier multiplier with polyhomogeneous symbol $p\left(\xi \right)$, and we derive algebraic decay estimates in terms of weighted Sobolev norms. Our basic example is the celebrated Benjamin–Ono equation

$$\left(0.1\right)\left(\right|D|+c)u=u^2,c>0,$$ 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.