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Mathematics > Analysis of PDEs

arXiv:1412.0362 (math)
[Submitted on 1 Dec 2014]

Title:Functions operating on modulation spaces and nonlinear dispersive equations

Authors:Divyang G. Bhimani, P. K. Ratnakumar
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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 $M^{p,q}_s(\R^n)$.
Comments: 27 pages
Subjects: Analysis of PDEs (math.AP)
Cite as: arXiv:1412.0362 [math.AP]
  (or arXiv:1412.0362v1 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.1412.0362

Submission history

From: Divyang Bhimani [view email]
[v1] Mon, 1 Dec 2014 06:57:45 UTC (23 KB)
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