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Evolution of the radius of spatial analyticity for the periodic BBM equation


Authors: A. Alexandrou Himonas and Gerson Petronilho
Journal: Proc. Amer. Math. Soc. 148 (2020), 2953-2967
MSC (2010): Primary 35Q53, 37K10
DOI: https://doi.org/10.1090/proc/14942
Published electronically: February 26, 2020
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Abstract: The Cauchy problem of the Benjamin-Bona-Mahony (BBM) equation with initial data $ u_0$ that are analytic on the torus and have uniform radius of analyticity $ r_0$ is considered, and the evolution of the radius of spatial analyticity $ r(t)$ of the solution $ u(t)$ at any future time $ t$ is examined. It is shown that the size of the radius of spatial analyticity persists for some time and after that it evolves in a such a way that its size at any time $ t$ is bounded below by $ c t^{-1}$ for some $ c>0$. The optimality of this bound remains an open question.


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Additional Information

A. Alexandrou Himonas
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: himonas.1@nd.edu

Gerson Petronilho
Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP 13565-905, Brazil
Email: gerson@dm.ufscar.br

DOI: https://doi.org/10.1090/proc/14942
Keywords: Periodic BBM equation, Cauchy problem, analytic spaces, uniform radius of analyticity, algebraic decrease, approximate conservation law.
Received by editor(s): January 14, 2019
Received by editor(s) in revised form: November 16, 2019
Published electronically: February 26, 2020
Additional Notes: The first author was partially supported by a grant from the Simons Foundation (#524469)
The second author was partially supported by grant 303111/2015-1, CNPq, and grant 2012/03168-7, São Paulo Research Foundation (FAPESP)
Communicated by: Catherine Sulem
Article copyright: © Copyright 2020 American Mathematical Society