J.evol.equ. 5 (2005) 01 – 33 1424–3199/05/010001 – 33 DOI 10.1007/s00028-004-0137-z © Birkh¨ auser Verlag, Basel, 2005 Quasi-linear parabolic problems in L 1 with non homogeneous conditions on the boundary Kaouther Ammar and Petra Wittbold Abstract. We are interested in parabolic problems with L 1 data of the type (P i,j (φ,ψ,β))        δ i u ′ (t) - div a(., Du) = φ(t) in Q := (0,T) × , δ j u ′ (t) + a(., Du).η + β(u) ∋ ψ(t) on  := (0,T) × ∂ δ i u(0, .) = u 0 in , δ j u(0, .) = u 0 on ∂, with i, j = 0, 1, (i, j )  = (0, 0), δ 0 = 0 and δ 1 = 1. Here,  is an open bounded subset of R N with regular boundary ∂ and a:  × R N → R N is a Caratheodory function satisfying the classical Leray-Lions conditions and β is a monotone graph in R 2 with closed domain and such that 0 ∈ β(0). We study these evolution problems from the point of view of semi-group theory, then we identify the generalized solution of the associated Cauchy problem with the entropy solution of (P i,j (φ,ψ,β)) in the usual sense introduced in [5]. 1. Introduction This paper is devoted to the solvability of evolution problems of the type (P i,j (φ,ψ,β))        δ i u ′ (t) - div a(., Du) = φ(t) in Q := (0,T) × , δ j u ′ (t) + a(., Du).η + β(u) ∋ ψ(t) on  := (0,T) × ∂ δ i u(0, .) = u 0 in , δ j u(0, .) = u 0 on ∂, i, j = 0, 1, (i, j )  = (0, 0), δ 0 = 0 and δ 1 = 1, where  is a bounded domain in R N ,ψ ∈ L 1 (0,T,L 1 (∂)), φ ∈ L 1 (0,T,L 1 ()), u 0 ∈ L 1 () and u 0 ∈ L 1 (∂). We suppose that β is a maximal monotone graph in R 2 with closed domain such that 0 ∈ β(0), and a:  × R N → R N is a Caratheodory function satisfying the classical Leray-Lions conditions, i.e. Received: 15 May 2003; accepted: 13 May 2004. Mathematics Subject Classification 2000: 35K55, 35K60, 35D05. Key words: operator, non local, accretive, entropy solution, “pseudo-envelope” solution.