Functions operating on modulation spaces and nonlinear dispersive equations

TL;DR

This paper characterizes functions that operate on modulation spaces and investigates the well-posedness of nonlinear dispersive equations with real analytic nonlinearities.

Contribution

It proves that functions operating on certain modulation spaces must be real analytic and characterizes these functions, also studying well-posedness of nonlinear dispersive equations.

Findings

Functions operating on $M^{p,1}$ are real analytic.

Characterization of functions operating on $M^{1,1}$.

Well-posedness results for NLS, NLW, NLKG with real entire nonlinearities.

Abstract

The aim of this paper is two fold. We show that if a complex function $F$ on $\C$ operates in the modulation spaces $M^{p,1}(\R^n)$ by composition, then $F$ is real analytic on $\R^2 \approx \C$. This answers negatively, the open question posed in [M. Ruzhansky, M. Sugimoto, B. Wang, Modulation Spaces and Nonlinear Evolution Equations, arXiv:1203.4651], regarding the general power type nonlinearity of the form $|u|^\alpha u$. We also characterise the functions that operate in the modulation space $M^{1,1}(\R^n)$. The local well-posedness of the NLS, NLW and NLKG equations for the `real entire' nonlinearities are also studied in some weighted modulation spaces $M^{p,q}_s(\R^n)$.