On a class of nonlinear parabolic equations with natural growth in non-reflexive Musielak spaces

An existence result of renormalized solutions for nonlinear parabolic Cauchy-Dirichlet problems whose model ⎧⎩⎨⎪⎪⎪⎪⎪⎪∂b(x,u)∂t−divA(x,t,u,∇u)−divΦ(x,t,u)=fb(x,u)(t=0)=b(x,u0)u=0 in Ω×(0,T) in Ω on ∂Ω×(0,T). is given in the non reflexive Musielak spaces, where b(x,⋅) is a strictly increasing C1-function for every x∈Ω with b(x,0)=0, the lower order term Φ is a non coercive Carath'{e}odory function satisfying only a natural growth condition described by the appropriate Musielak function φ and f is an integrable data.