Illposedness for dispersive equations: Degenerate dispersion and Takeuchi--Mizohata condition

We provide a unified viewpoint on two illposedness 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-magnetohydynamics. Concretely, the main results in this paper concern strong illposedness of the Cauchy problem (e.g., non-existence 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, $K(m,n)$ models of Rosenau–Hyman, and the inviscid surface growth model. The mechanism behind these results may be understood in terms of 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 $L^2$-illposedness result by Mizohata for linear variable-coefficient Schrödinger equations with failed Takeuchi–Mizohata condition.