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

Akhunov, Timur

doi:10.1090/S0002-9939-2014-12136-8

A sharp condition for the well-posedness of the linear KdV-type equation
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Proc. Amer. Math. Soc. 142 (2014), 4207-4220 Request permission

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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