Parabolic equations in anisotropic orlicz spaces with general N-functions

Abstract

In the present paper we study the existence of weak solutions to an abstract parabolic initial-boundary value problem. On the operator appearing in the equation we assume the coercivity conditions given by an N-function (i.e., convex function satisfying conditions specified in the paper). The main novelty of the paper consists in the lack of any growth restrictions on the N-function combined with an anisotropic character of the N-function, namely we allow the dependence on all the directions of the gradient, not only on its absolute value.