Geodesic
Existence of entropy measure-valued solutions for forward-backward $p$-parabolic equations
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 774-793

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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. 
@article{SEMR_2017_14_a93,
     author = {S. N. Antontsev and I. V. Kuznetsov},
     title = {Existence of entropy measure-valued solutions for forward-backward $p$-parabolic equations},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {774--793},
     publisher = {mathdoc},
     volume = {14},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2017_14_a93/}
}
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S. N. Antontsev; I. V. Kuznetsov. Existence of entropy measure-valued solutions for forward-backward $p$-parabolic equations. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 774-793. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a93/