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 differentiation. 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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