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Title: ILL-POSEDNESS FOR DISPERSIVE EQUATIONS: DEGENERATE DISPERSION AND THE TAKEUCHII-MIZOHATA CONDITION

Abstract: We provide a unified viewpoint on two ill-posedness mechanisms for dispersive equations in one spatial dimension, namely degenerate dispersion and (the failure of) the Takeuchi-Mizohata condition. Our approach is based on a robust energy-and duality-based method introduced in an earlier work of the authors in the setting of Hall-magnetohydrodynamics. Concretely, the main results in this paper concern strong ill-posedness of the Cauchy problem (e.g., nonexistence and unboundedness of the solution map) in high-regularity Sobolev spaces for various quasilinear degenerate Schr & ouml;dinger-and KdV-type equations, including the Hunter-Smothers equation, K (m, n) models of Rosenau-Hyman, and the inviscid surface growth model. The mechanism behind these results may be understood in terms of the combination of two effects: degenerate dispersion-which is a property of the principal term in the presence of degenerating coefficients - and the evolution of the amplitude governed by the Takeuchi-Mizohata condition - which concerns the subprincipal term. We also demonstrate how the same techniques yield a more quantitative version of the classical L2-ill-posedness result by Mizohata for linear variable-coefficient Schr & ouml;dinger equations with failed Takeuchi-Mizohata condition.

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