Abstract:
The problem of the absence of global solutions of initial-boundary value problems for the
Kuramoto-Sivashinsky equation is considered. Sufficient conditions for the absence of global solutions of the problems under consideration are obtained both for bounded and unbounded
domains. These conditions imply a priori the blow-up of the solution of the corresponding initial-boundary value problem. The proof uses a generalization of the method of non-linear capacity based on the choice of asymptotically optimal test functions.
Bibliography: 20 titles.
\Bibitem{Pok08}
\by S.~I.~Pokhozhaev
\paper On blow-up of solutions of the Kuramoto-Sivashinsky equation
\jour Sb. Math.
\yr 2008
\vol 199
\issue 9
\pages 1355--1365
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This publication is cited in the following 15 articles:
Matthew Enlow, Adam Larios, Jiahong Wu, “Algebraic calming for the 2D Kuramoto-Sivashinsky equations”, Nonlinearity, 37:11 (2024), 115019
M. O. Korpusov, A. Yu. Perlov, A. V. Tymoshenko, R. S. Shafir, “On the Blow-Up of the Solution of a Nonlinear System of Equations of a Thermal-Electrical Model”, Math. Notes, 114:5 (2023), 850–861
Vo Van Au, “Analysis of large time asymptotics of the fourth‐order parabolic system involving variable coefficients and mixed nonlinearities”, Math Methods in App Sciences, 46:14 (2023), 15305
Adam Larios, Mohammad Mahabubur Rahman, Kazuo Yamazaki, “Regularity Criteria for the Kuramoto–Sivashinsky Equation in Dimensions Two and Three”, J Nonlinear Sci, 32:6 (2022)
Larios A., Yamazaki K., “On the Well-Posedness of An Anisotropically-Reduced Two-Dimensional Kuramoto-Sivashinsky Equation”, Physica D, 411 (2020), 132560
M. O. Korpusov, “The finite-time blowup of the solution of an initial boundary-value
problem for the nonlinear equation of ion sound waves”, Theoret. and Math. Phys., 187:3 (2016), 835–841
Bilgin B., Kalantarov V., Zelik S., “Preventing Blow up by Convective Terms in Dissipative PDE's”, J. Math. Fluid Mech., 18:3 (2016), 463–479
M. O. Korpusov, “On the Blow-Up of the Solution of an Equation Related to the Hamilton–Jacobi Equation”, Math. Notes, 93:1 (2013), 90–101
A. Eden, V. K. Kalantarov, S. V. Zelik, “Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials”, J. Math. Phys., 54:4 (2013), 041502
Pokhozhaev S.I., “Critical Nonlinearities in Partial Differential Equations”, Russ. J. Math. Phys., 20:4 (2013), 476–491
M. O. Korpusov, “Blowup of solutions of the three-dimensional Rosenau–Burgers equation”, Theoret. and Math. Phys., 170:3 (2012), 280–286
Korpusov M.O., “On the blow-up of solutions of the Benjamin-Bona-Mahony-Burgers and Rosenau-Burgers equations”, Nonlinear Anal., 75:4 (2012), 1737–1743
Korpusov M.O., “On the blow-up of the solution of an equation with a gradient nonlinearity”, Differ. Equ., 48:6 (2012), 796–808
A. N. Bogolyubov, M. D. Malykh, “On a class of nonlocal parabolic equations”, Comput. Math. Math. Phys., 51:6 (2011), 987–993
Pohozaev S.I., “Critical nonlinearities in partial differential equations”, Milan J. Math., 77:1 (2009), 127–150
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