Bol. Soc. Paran. Mat. (3s.) v. 38 6 (2020): 99–126. c SPM –ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v38i6.36594 Existence of Solutions for Some Strongly Nonlinear Parabolic Problems Involving Lower Order Terms in Divergence Form in Musielak-Orlicz Spaces Abdeslam Talha and Abdelmoujib Benkirane abstract: In this paper, we study an existence of solutions for a class of non- linear parabolic problems with two lower order terms and L 1 -data in the context of Musielak-Orlicz spaces. Key Words: Parabolic problems, Inhomogeneous Musielak-Orlicz-Sobolev space, Lower order term, Truncation. Contents 1 Introduction: 99 2 Preliminaries 100 2.1 Musielak-Orlicz functions ........................ 100 2.2 Musielak-Orlicz spaces .......................... 101 2.3 Inhomogeneous Musielak-Orlicz-Sobolev spaces ............ 104 3 Essential Assumptions 105 4 Some technical Lemmas 106 5 Main results 108 1. Introduction: Let Ω be a bounded open set of R N , T is a positive real number, and Q = Ω × [0,T ]. We deal with boundary value problem: (P)      ∂u ∂t − div ( a(x,t,u, ∇u) + Φ(x,t,u) ) + g(x,t,u, ∇u)= f in Q, u =0 in ∂ Ω × (0,T ). u(x, 0) = u 0 in Ω, where A(u)= −div(a(x,t,u, ∇u)) is a Leray-Lions Operator defined on D(A) ⊂ W 1,x 0 L ϕ (Q) −→ W −1,x L ψ (Q) where ϕ and ψ are two complementary Musielak- Orlicz functions. The lower order term Φ : Ω × (0,T ) × R −→ R N is a Carath´ eodory function satisfies the following growth condition for a.e. (x,t) ∈ Q and for all s ∈ R, |Φ(x,t,s)|≤ P (x,t) γ x −1 γ x (|s|). 2010 Mathematics Subject Classification: 46E35, 35K15, 35K20, 35K60. Submitted April 06, 2017. Published November 10, 2017 99 Typeset by B S P M style. c  Soc. Paran. de Mat.