Decay estimates for solutions of nonlocal semilinear equations

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.