Abstract:
We consider a series of initial-boundary value problems for the equation
of ion-sound waves in a plasma. For each of them we prove the local (in time)
solubility and perform an analytical-numerical study of the blow-up of solutions.
We use the method of test functions to obtain sufficient conditions for
finite-time blow-up and an upper bound for the blow-up time. In concrete
numerical examples we improve these bounds numerically using the mesh refinement
method. Thus the analytical and numerical parts of the investigation complement
each other. The time interval for the numerical modelling is chosen
in accordance with the analytically obtained upper bound for the blow-up time.
In return, numerical calculations specify the moment and pattern of this blow-up.
Keywords:
blow-up of a solution, non-linear initial-boundary value problem,
Sobolev-type equations, exponential non-linearity, Richardson extrapolation.
This paper was written with the support of the Russian Foundation
for Basic Research (grants nos. 15-01-03524, 16-32-00011, 16-01-00755,
16-01-00437, 14-01-00182 and 14-01-00208).
Citation:
M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, E. V. Yushkov, “Blow-up of solutions of a full non-linear equation of ion-sound waves
in a plasma with non-coercive non-linearities”, Izv. Math., 82:2 (2018), 283–317
\Bibitem{KorLukPan18}
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\paper Blow-up of solutions of a~full non-linear equation of ion-sound waves
in a~plasma with non-coercive non-linearities
\jour Izv. Math.
\yr 2018
\vol 82
\issue 2
\pages 283--317
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Linking options:
https://www.mathnet.ru/eng/im8579
https://doi.org/10.1070/IM8579
https://www.mathnet.ru/eng/im/v82/i2/p43
This publication is cited in the following 11 articles:
M. O. Korpusov, E. A. Ovsyannikov, “Local solvability, blow-up, and Hölder regularity of solutions to some Cauchy problems for nonlinear plasma wave equations: I. Green formulas”, Comput. Math. and Math. Phys., 62:10 (2022), 1609–1631
M. O. Korpusov, A. A. Panin, A. E. Shishkov, “On the critical exponent “instantaneous blow-up” versus “local solubility” in the Cauchy problem for a model equation of Sobolev type”, Izv. Math., 85:1 (2021), 111–144
M. O. Korpusov, “Nonlinear Equations of the Theory of Ion-Sound Plasma Waves”, Comput. Math. and Math. Phys., 61:11 (2021), 1886
A. A. Panin, G. I. Shlyapugin, “Local Solvability and Global Unsolvability of a Model of Ion-Sound Waves in a Plasma”, Math. Notes, 107:3 (2020), 464–477
M. O. Korpusov, E. A. Ovsyannikov, “Blow-up instability in non-linear wave models with distributed parameters”, Izv. Math., 84:3 (2020), 449–501
M. O. Korpusov, “Blow-up and global solubility in the classical sense of the Cauchy problem for a formally hyperbolic equation with a non-coercive source”, Izv. Math., 84:5 (2020), 930–959
M. O. Korpusov, A. N. Levashov, D. V. Lukyanenko, “Analytical-numerical study of finite-time blow-up of the solution to the initial-boundary value problem for the nonlinear Klein–Gordon equation”, Comput. Math. Math. Phys., 60:9 (2020), 1452–1460
I. I. Kolotov, A. A. Panin, “On Nonextendable Solutions and Blow-Ups of Solutions of Pseudoparabolic Equations with Coercive and Constant-Sign Nonlinearities: Analytical and Numerical Study”, Math. Notes, 105:5 (2019), 694–706
M. O. Korpusov, A. K. Matveeva, D. V. Lukyanenko, “Diagnostika mgnovennogo razrusheniya resheniya v nelineinom uravnenii teorii voln v poluprovodnikakh”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 12:4 (2019), 104–113
Dmitry Lukyanenko, Nikolay Nefedov, Lecture Notes in Computer Science, 11386, Finite Difference Methods. Theory and Applications, 2019, 72
M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, G. I. Shlyapugin, “On the blow-up phenomena for a 1-dimensional equation of ion sound waves in a plasma: analytical and numerical investigation”, Math. Methods Appl. Sci., 41:8 (2018), 2906–2929
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