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arXiv · 1203.2075

Decay estimates for solutions of nonlocal semilinear equations

Marco Cappiello·Todor Gramchev·Luigi Rodino
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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.

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Marco Cappiello, Todor Gramchev, Luigi Rodino. 2012-03-09. Decay estimates for solutions of nonlocal semilinear equations. https://arxiv.org/abs/1203.2075

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