J. Nonl. Evol. Equ. Appl. 2013 (6), pp. 67-87, published on August 20, 2014:

Unique solvability of initial boundary value problems for anisotropic elliptic-parabolic equations with variable exponents of nonlinearity

M. M. Bokalo, O. M. Buhrii

Department of Differential Equations, Faculty of Mechanics and Mathematics, Ivan Franko National University of Lviv ; 79000-Lviv, Ukraine

R. A. Mashiyev

Department of Mathematics, Faculty of Science, Dicle University ; 21280-Diyarbakir, Turkey

Received on November 13, 2012, revised version on January 17, 2013
Accepted on January 31, 2013

Communicated by Alexander Pankov

Abstract.  Existence and uniqueness of weak solutions of initial-boundary-value problems for second order elliptic-parabolic equations are proved.These equations have the exponents of nonlinearity depending on the points of domain and the direction of diļ¬€erentiation. The weak solutions belong to some generalized Sobolev spaces.
Keywords: nonlinear equation, anisotropic equation, elliptic-parabolic equation, degenerate parabolic equation, initial-boundary-value problem, variable exponents of nonlinearity, generalized Lebesgue space, generalized Sobolev space.
2010 Mathematics Subject Classification:   35D05, 35J25, 35J60, 35K15, 35K55.

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