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 \begin{equation} \label{BO}(|D|+c)u=u^2, \qquad c>0,\end{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.

Decay estimates for solutions of nonlocal semilinear equations

CAPPIELLO, Marco;RODINO, Luigi Giacomo
2015

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 \begin{equation} \label{BO}(|D|+c)u=u^2, \qquad c>0,\end{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.
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http://projecteuclid.org/download/pdf_1/euclid.nmj/1431347891
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
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