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 algebraic decay estimates in terms of weighted Sobolev norms. Our basic example is the celebrated Benjamin-Ono equation $$\label{BO}(|D|+c)u=u^2, \qquad c>0,$$ 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.

### Decay estimates for solutions of nonlocal semilinear equations

#### 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 algebraic decay estimates in terms of weighted Sobolev norms. Our basic example is the celebrated Benjamin-Ono equation $$\label{BO}(|D|+c)u=u^2, \qquad c>0,$$ 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.
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Decay estimates, nonlocal semilinear elliptic equations, solitary waves
Cappiello, Marco; Gramchev, Todor; Rodino, Luigi
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2318/1548079