Fifth-order dispersive equations arise in the context of higher-order models for phenomena such as water waves. For fifth-order variable-coefficient linear dispersive equations, we provide conditions under which the intitial value problem is either well-posed or ill-posed. For well-posedness, a balance must be struck between the leading-order dispersion and possible backwards diffusion from the fourth-derivative term. This generalizes work by the first author and Wright for third-order equations. In addition to inherent interest in fifth-order dispersive equations, this work is also motivated by a question from numerical analysis: finite difference schemes for third-order numerical equations can yield approximate solutions which effectively satisfy fifth-order equations. We find that such a fifth-order equation is well-posed if and only if the underlying third-order equation is ill-posed.

在水波等现象的高阶模型中，出现了五阶色散方程。对于五阶变系数线性色散方程，我们提供了条件条件下初始值问题是适定的还是不适定的条件。为了具有良好的适度性，必须在前导色散与四阶导数项可能的向后扩散之间取得平衡。这归纳了第一作者和赖特对三阶方程的研究。除了对五阶色散方程固有的兴趣外，这项工作还受到数值分析问题的推动：三阶数值方程的有限差分方案可以产生有效满足五阶方程的近似解。