Non-coercive nonlinear parabolic problem with measurableobstacle in modular space

Abstract

In this paper, we shall be concerned with the existence result for an unilateral problem associated to the form, ∂u ∂t −div ( a(x, t, u, ∇u) ) −div(Φ(x, t, u))) = f in QT = Ω× (0, T ), where Au = −diva(x, t, u, ∇u) is a Leray-Lions operator defined on W 1,x 0 LM (QT ), The growth and coercivity conditions on the monotone vector field a are prescribed by generalized N -function M , which does not have to satisfy the ∆2 condition. Therefore we work with a generalized Musielak-Orlicz spaces which are not necessarily reflexive. The right hand side and the initial data belong to L1(QT ) and the lower order term Φ is a non-coercive Carath ́eodory function.