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A note on the Kuramoto-Sivashinsky equation with discontinuity

  • Received: 25 July 2020 Accepted: 09 September 2020 Published: 22 October 2020
  • In this work we consider differential equations of the type $$\pm\, u^{(k)} = f(u), $$ and study the extinction profile of their solutions. Emphasis is placed on the special case $-u^{(4)} = sgn(u)$, which is related to the Kuramoto-Sivashinsky equation. In this case we describe in more detail the extinction phenomenon and prove a conjecture by Galaktionov and Svirshchevskii.

    Citation: Lorenzo D'Ambrosio, Marco Gallo, Alessandro Pugliese. A note on the Kuramoto-Sivashinsky equation with discontinuity[J]. Mathematics in Engineering, 2021, 3(5): 1-29. doi: 10.3934/mine.2021041

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  • In this work we consider differential equations of the type $$\pm\, u^{(k)} = f(u), $$ and study the extinction profile of their solutions. Emphasis is placed on the special case $-u^{(4)} = sgn(u)$, which is related to the Kuramoto-Sivashinsky equation. In this case we describe in more detail the extinction phenomenon and prove a conjecture by Galaktionov and Svirshchevskii.
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    © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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