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 Sib. Èlektron. Mat. Izv., 2017, Volume 14, Pages 774–793 (Mi semr823)

Differentical equations, dynamical systems and optimal control

Existence of entropy measure-valued solutions for forward-backward $p$-parabolic equations

S. N. Antontsevabc, I. V. Kuznetsovba

a Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, pr. Acad. Lavrentyeva 15, 630090, Novosibirsk, Russia
b Novosibirsk State University, Pirogova st., 2, 630090, Novosibirsk, Russia
c CMAF-CIO, University of Lisbon, 1749-016 Lisbon, Portugal

Abstract: In this paper we have proved that the Dirichlet problem for the forward-backward $p$-parabolic equation has an entropy measure-valued solution which has been obtained as a singular limit of weak solutions and their gradients to the Dirichlet problem for the elliptic equation containing the anisotropic $(p,2)$-Laplace operator. In order to guarantee the existence of entropy measure-valued solutions, the initial and final conditions should be formulated in the form of integral inequalities. This means that an entropy measure-valued solution can deviate from both initial and final data on the boundary. Moreover, a gradient Young measure appears in the representation of an entropy measure-valued solution. The uniqueness of entropy measure-valued solutions is still an open question.

Keywords: anisotropic Laplace operator, entropy measure-valued solution, forward-backward parabolic equation, gradient Young measure.

 Funding Agency Grant Number Russian Science Foundation 15-11-20019 The work was supported by the Russian Science Foundation (Grant 15-11-20019).

DOI: https://doi.org/10.17377/semi.2017.14.066

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UDC: 517.95
MSC: 35K92
Received May 28, 2017, published August 16, 2017
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Citation: S. N. Antontsev, I. V. Kuznetsov, “Existence of entropy measure-valued solutions for forward-backward $p$-parabolic equations”, Sib. Èlektron. Mat. Izv., 14 (2017), 774–793

Citation in format AMSBIB
\Bibitem{AntKuz17} \by S.~N.~Antontsev, I.~V.~Kuznetsov \paper Existence of entropy measure-valued solutions for forward-backward $p$-parabolic equations \jour Sib. \Elektron. Mat. Izv. \yr 2017 \vol 14 \pages 774--793 \mathnet{http://mi.mathnet.ru/semr823} \crossref{https://doi.org/10.17377/semi.2017.14.066} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000407792200066} `

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This publication is cited in the following articles:
1. I. V. Kuznetsov, S. A. Sazhenkov, “Anisotropic vanishing diffusion method applied to genuinely nonlinear forward-backward ultra-parabolic equations”, Sib. elektron. matem. izv., 15 (2018), 1158–1173
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