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Mathematics in Engineering

2021, Volume 3, Issue 5: 1-29. doi: 10.3934/mine.2021041
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A note on the Kuramoto-Sivashinsky equation with discontinuity

  • Dipartimento di Matematica, Università degli Studi di Bari, via E. Orabona, 4, I-70125 Bari, Italy
  • 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

    Related Papers:

  • 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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    [1] Acary V, Brogliato B (2008) Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics, Berlin, Heidelberg: Springer.
    [2] Alama YB, Lessard JP (2020) Traveling wave oscillatory patterns in a signed Kuramoto-Sivashinsky equation with absorption. J Comput Appl Math 372: 1-7.
    [3] Bernardo M, Budd C, Champneys AR, et al. (2008) Piecewise-Smooth Dynamical Systems: Theory and Applications, London: Springer.
    [4] Bernis F, McLeod JB (1991) Similarity solutions of a higher order nonlinear diffusion equation. Nonlinear Anal 17: 1039-1068.
    [5] D'Ambrosio L, Lessard JP, Pugliese A (2015) Blow-up profile for solutions of a fourth order nonlinear equation. Nonlinear Anal 121: 280-335.
    [6] D'Ambrosio L, Mitidieri E, Liouville theorems for a semilinear biharmonic equation. Preprint.
    [7] Evans JD, Galaktionov VA, King JR (2007) Source-type solutions of the fourth-order unstable thin film equation. European J Appl Math 18: 273-321.
    [8] Filippov AF (2013) Differential Equations with Discontinuous Righthand Sides, Springer Science & Business Media.
    [9] Galaktionov VA, Mitidieri E, Pohozaev SI (2014) Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrodinger Equations, CRC Press.
    [10] Galaktionov VA, Svirshchevskii SR (2006) Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, CRC Press.
    [11] Gallo M (2016) Results on some nonlinear fourth order differential equations, Undergraduate's thesis, University of Bari.
    [12] Gouze JL, Sari T (2010) A class of piecewise linear differential equations arising in biological models. Dyn Syst 17: 299-316.
    [13] Leine R, Nijmeijer H (2004) Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Berlin: Springer.
    [14] Makarenkov O, Lamb JSW (2012) Dynamics and bifurcations of nonsmooth systems: A survey. Phys D 241: 1826-1844.
    [15] Utkin VI (1992) Sliding Modes in Control and Optimization, Berlin: Springer.
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