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 Mat. Sb., 1989, Volume 180, Number 9, Pages 1183–1210 (Mi msb1656)

The Cauchy problem for odd-order quasilinear equations

A. V. Faminskii

Abstract: A nonlocal Cauchy problem for multidimensional quasilinear evolution equations containing a linear differential operator $L(t,x,D_x)$ with leading derivatives of odd order is considered. The conditions on the nonlinear terms are chosen so that they are subordinate to the operator $L$. The Korteweg–de Vries equation is a special case of such equations. No smoothness conditions are imposed on the initial function $u_0(x)$ $(u_0(x)\in L_2(\mathbf R^n))$. Theorems on the existence, uniqueness, and continuous dependence on the initial data of generalized solutions are established.
Bibliography: 20 titles.

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English version:
Mathematics of the USSR-Sbornik, 1991, 68:1, 31–59

Bibliographic databases:

UDC: 517.9
MSC: Primary 35K22; Secondary 35Q20, 35D05

Citation: A. V. Faminskii, “The Cauchy problem for odd-order quasilinear equations”, Mat. Sb., 180:9 (1989), 1183–1210; Math. USSR-Sb., 68:1 (1991), 31–59

Citation in format AMSBIB
\Bibitem{Fam89} \by A.~V.~Faminskii \paper The Cauchy problem for odd-order quasilinear equations \jour Mat. Sb. \yr 1989 \vol 180 \issue 9 \pages 1183--1210 \mathnet{http://mi.mathnet.ru/msb1656} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1017821} \zmath{https://zbmath.org/?q=an:0701.35046|0712.35085} \transl \jour Math. USSR-Sb. \yr 1991 \vol 68 \issue 1 \pages 31--59 \crossref{https://doi.org/10.1070/SM1991v068n01ABEH001932} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1991EX22700003} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. K. Sangare, A. V. Faminskii, “Weak Solutions of a Mixed Problem in a Half-Strip for a Generalized Kawahara Equation”, Math. Notes, 85:1 (2009), 90–100
2. Kuvshinov R.V., Faminskii A.V., “Mixed problem for the Kawahara equation in a half-strip”, Differ. Equ., 45:3 (2009), 404–415
3. Larkin N.A., Luchesi J., “General Mixed Problems for the KdV Equations on Bounded Intervals”, Electron. J. Differ. Equ., 2010, 168
4. Faminskii A.V., Larkin N.A., “Initial-Boundary Value Problems for Quasilinear Dispersive Equations Posed on a Bounded Interval”, Electron. J. Differ. Equ., 2010, 01
5. Faminskii A.V., “Weak Solutions to Initial-Boundary-Value Problems for Quasilinear Evolution Equations of an Odd Order”, Adv. Differ. Equat., 17:5-6 (2012), 421–470
6. A. V. Faminskii, M. A. Opritova, “On the initial-value problem for the Kawahara equation”, Journal of Mathematical Sciences, 201:5 (2014), 614–633
7. Larkin N.A., Simoes M.H., “General Boundary Conditions for the Kawahara Equation on Bounded Intervals”, Electron. J. Differ. Equ., 2013, 159
8. Baykova E.S., Faminskii A.V., “On Initial-Boundary-Value Problems in a Strip for the Generalized Two-Dimensional Zakharov-Kuznetsov Equation”, Adv. Differ. Equat., 18:7-8 (2013), 663–686
9. A. P. Antonova, A. V. Faminskii, “On the Regularity of Solutions of the Cauchy Problem for the Zakharov–Kuznetsov Equation in Hölder Norms”, Math. Notes, 97:1 (2015), 12–20
10. Faminskii A.V., “Initial-Boundary Value Problems in a Rectangle For Two-Dimensional Zakharov-Kuznetsov Equation”, J. Math. Anal. Appl., 463:2 (2018), 760–793
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