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Well-posedness and Ill-posedness for Linear Fifth-Order Dispersive Equations in the Presence of Backwards Diffusion
Journal article   Peer reviewed

Well-posedness and Ill-posedness for Linear Fifth-Order Dispersive Equations in the Presence of Backwards Diffusion

David M. Ambrose and Jacob Woods
Journal of dynamics and differential equations, v 34(2), pp 897-917
31 Oct 2020
DOI: https://doi.org/10.1007/s10884-020-09905-9
Featured in Collection :   UN Sustainable Development Goals @ Drexel

Abstract

Mathematics Mathematics, Applied Physical Sciences Science & Technology
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.

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Collaboration types
Domestic collaboration
Web of Science research areas
Mathematics
Mathematics, Applied
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