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A sharp condition for the well-posedness of the linear KdV-type equation


Author: Timur Akhunov
Journal: Proc. Amer. Math. Soc. 142 (2014), 4207-4220
MSC (2010): Primary 35Q53
DOI: https://doi.org/10.1090/S0002-9939-2014-12136-8
Published electronically: August 7, 2014
MathSciNet review: 3266990
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Abstract: An initial value problem for a very general linear equation of KdV-type is considered. Assuming non-degeneracy of the third derivative coefficient, this problem is shown to be well-posed under a certain simple condition, which is an adaptation of the Mizohata-type condition from the Schrödinger equation to the context of KdV. When this condition is violated, ill-posedness is shown by an explicit construction. These results justify formal heuristics associated with dispersive problems and have applications to non-linear problems of KdV-type.


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Additional Information

Timur Akhunov
Affiliation: Department of Mathematics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada
Address at time of publication: Department of Mathematics, 820 Hylan Building, University of Rochester, Rochester, New York 14627
Email: takhunov@ur.rochester.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12136-8
Keywords: KdV, linear, dispersive, partial differential equations, energy method, Mizohata condition
Received by editor(s): October 11, 2012
Received by editor(s) in revised form: January 9, 2013
Published electronically: August 7, 2014
Communicated by: Joachim Krieger
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.