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
Schrödinger- and KdV-type equations, including the Hunter–Smothers equation,
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
-ill-posedness
result by Mizohata for linear variable-coefficient Schrödinger equations with failed
Takeuchi–Mizohata condition.
