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On the asymptotic behavior of solutions to the Benjamin-Ono equation


Authors: Claudio Muñoz and Gustavo Ponce
Journal: Proc. Amer. Math. Soc. 147 (2019), 5303-5312
MSC (2010): Primary 37K15, 35Q53; Secondary 35Q51, 37K10
DOI: https://doi.org/10.1090/proc/14643
Published electronically: June 10, 2019
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Abstract: We prove that the limit infimum, as time $ \,t\,$ goes to infinity, of any uniformly bounded in time $ H^1\cap L^1$ solution to the Benjamin-Ono equation converge to zero locally in an increasing in time region of space of order $ \,t/\log t$. Also for a solution with a mild $ L^1$-norm growth in time, its limit infimum must converge to zero, as time goes to infinity, locally in an increasing on time region of space of order depending of the rate of growth of its $ L^1$-norm. In particular, we discard the existence of breathers and other solutions for the BO model moving with a speed ``slower'' than a soliton.


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Additional Information

Claudio Muñoz
Affiliation: CNRS and Departamento de Ingeniería Matemática DIM-CMM UMI 2807-CNRS, Universidad de Chile, Santiago, Chile
Email: cmunoz@dim.uchile.cl

Gustavo Ponce
Affiliation: Department of Mathematics, University of California-Santa Barbara, Santa Barbara, California 93106
Email: ponce@math.ucsb.edu

DOI: https://doi.org/10.1090/proc/14643
Keywords: BO equations, decay estimates
Received by editor(s): October 4, 2018
Received by editor(s) in revised form: March 27, 2019
Published electronically: June 10, 2019
Communicated by: Catherine Sulem
Article copyright: © Copyright 2019 American Mathematical Society