Renormalized solutions to nonlinear parabolic problems in generalized Musielak–Orlicz spaces

Abstract

We will present the proof of existence of renormalized solutions to a nonlinear parabolic problem ∂tu−diva(⋅,Du)=f with right-hand side f and initial data u0 in L1. The growth and coercivity conditions on the monotone vector field a are prescribed by a generalized N-function M which is anisotropic and inhomogeneous with respect to the space variable. In particular, M does not have to satisfy an upper growth bound described by a Δ2-condition. Therefore we work with generalized Musielak–Orlicz spaces which are not necessarily reflexive. Moreover we provide a weak sequential stability result for a more general problem: ∂tβ(⋅,u)−div(a(⋅,Du)+F(u))=f, where β is a monotone function with respect to the second variable and F is locally Lipschitz continuous. Within the proof we use truncation methods, Young measure techniques, the integration by parts formula and monotonicity arguments which have been adapted to nonreflexive Musielak–Orlicz spaces.