Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations

Marco  Cappiello; Fabio Nicola

doi:10.3934/dcds.2016.36.1869

Article Contents

2016, Volume 36, Issue 4: 1869-1880. Doi: 10.3934/dcds.2016.36.1869

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Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations

Marco Cappiello^1, and
Fabio Nicola^2,

1.
Dipartimento di Matematica, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 - Torino
2.
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received: February 28, 2015

Revised: February 28, 2015

Published: May 2017

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Abstract

We consider semilinear equations of the form $p(D)u=F(u)$, with a locally bounded nonlinearity $F(u)$, and a linear part $p(D)$ given by a Fourier multiplier. The multiplier $p(\xi)$ is the sum of positively homogeneous terms, with at least one of them non smooth. This general class of equations includes most physical models for traveling waves in hydrodynamics, the Benjamin-Ono equation being a basic example.
We prove sharp pointwise decay estimates for the solutions to such equations, depending on the degree of the non smooth terms in $p(\xi)$. When the nonlinearity is smooth we prove similar estimates for the derivatives of the solution, as well as holomorphic extension to a strip, for analytic nonlinearity.

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Mathematics Subject Classification: Primary: 35J61; Secondary: 35B40, 35Q51.

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