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

We will present the proof of existence of renormalized solutions to a nonlinear parabolic problem ∂tu−diva(⋅,Du)=f with right-hand side ff and initial data u0u0 in L1L1. The growth and coercivity conditions on the monotone vector field aa are prescribed by a generalized NN-function MM which is anisotropic and inhomogeneous with respect to the space variable. In particular, MM does not have to satisfy an upper growth bound described by a Δ2Δ2-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 ββ is a monotone function with respect to the second variable and FF 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.