Functions operating on modulation spaces and nonlinear dispersive equations

Abstract

The aim of this paper is two fold. We show that if a complex function $F$ on $\C$ operates in the modulation spaces $M^{p,1}({\mathbb{R}}^n)$ 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 $M^{1,1}({\mathbb{R}}^n)$. The local well-posedness of the NLS, NLW and NLKG equations for the `real entire' nonlinearities are also studied in some weighted modulation spaces $M_s^{p,q}({\mathbb{R}}^n)$.

27 pages