Obstacle parabolic equations in non-reflexive Musielak-Orlicz spaces

Abstract

AbstractWe prove existence of entropy solutions to general class of unilateral nonlinear parabolic equation in inhomogeneous Musielak-Orlicz spaces avoiding ceorcivity restrictions on the second lower order term. Namely, we consider{u≥ψinQT,∂b(x,u)∂t-div(a(x,t,u,∇u))=f+div(g(x,t,u))∈L1(QT).$$\left\{ \matrix{ \matrix{ {u \ge \psi } \hfill & {{\rm{in}}} \hfill & {{Q_T},} \hfill \cr } \hfill \cr {{\partial b(x,u)} \over {\partial t}} - div\left( {a\left( {x,t,u,\nabla u} \right)} \right) = f + div\left( {g\left( {x,t,u} \right)} \right) \in {L^1}\left( {{Q_T}} \right). \hfill \cr} \right.$$The growths of the monotone vector fielda(x,t,u, ᐁu) and the non-coercive vector fieldg(x,t,u) are controlled by a generalized nonhomogeneousN- functionM(see (3.3)-(3.6)). The approach does not require any particular type of growth ofM(neither Δ2nor ᐁ2). The proof is based on penalization method.