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

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Abstract

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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Acknowledgements

The first author gratefully acknowledges support from the National Science Foundation through grants DMS-1515849 and DMS-1907684. The authors thank the Department of Mathematics of Drexel University for support for the second author to undertake this research as part of an undergraduate research co-op experience. The authors thank Gideon Simpson for helpful conversations, especially relating to effective equations for finite difference schemes. The authors also thank J. Douglas Wright for helpful conversations, especially relating to ill-posedness.

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  1. Jacob Woods

    Present address: Department of Mathematics, Temple University, Philadelphia, PA, USA

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  1. Department of Mathematics, Drexel University, Philadelphia, PA, USA

    David M. Ambrose & Jacob Woods

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  1. David M. Ambrose
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  2. Jacob Woods
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Correspondence to David M. Ambrose.

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Ambrose, D.M., Woods, J. Well-posedness and Ill-posedness for Linear Fifth-Order Dispersive Equations in the Presence of Backwards Diffusion. J Dyn Diff Equat 34, 897–917 (2022). https://doi.org/10.1007/s10884-020-09905-9

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  • DOI: https://doi.org/10.1007/s10884-020-09905-9

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