Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations

TL;DR

This paper establishes sharp decay estimates and smoothness properties for solutions to a broad class of nonlocal semilinear equations, including models like the Benjamin-Ono equation, highlighting the influence of non-smooth linear operators.

Contribution

It provides new sharp decay estimates for solutions and their derivatives, accounting for non-smooth Fourier multipliers in a general class of nonlocal semilinear equations.

Findings

Sharp pointwise decay estimates depending on non-smoothness of p(\xi)

Smoothness results for derivatives when nonlinearity is smooth

Holomorphic extension of solutions in the analytic case

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 holomorphic extension to a strip, for analytic nonlinearity.