J.evol.equ. 3 (2003) 407 – 418 1424–3199/03/030407 – 12 DOI 10.1007/s00028-003-0109-7 © Birkh¨ auser Verlag, Basel, 2003 Some noncoercive parabolic equations with lower order terms in divergence form Lucio Boccardo 1 , Luigi Orsina 2 and Alessio Porretta 3 Trois g´ en´ erations d’amis romains d´ edient cet article ` a Philippe Abstract. This paper deals with existence and regularity results for the problem    u t − div(a(x,t,u)∇u) =−div(uE) in  × (0,T), u = 0 on ∂ × (0,T), u(0) = u 0 in , under various assumptions on E and u 0 . The main difficulty in studying this problem is due to the presence of the term div(uE), which makes the differential operator non coercive on the “energy space” L 2 (0,T ; H 1 0 ()). 1. Introduction In this paper we are going to study the following parabolic problem    u t − div(a(x,t,u)∇u) =−div(uE) in  × (0,T), u = 0 on ∂ × (0,T), u(0) = u 0 in , (1) where  is a bounded open subset of R N , N ≥ 2 and T> 0. We assume that a(x,t,s) :  × (0,T) × R → R is a function which is continuous with respect to s for almost every (x,t) ∈  × (0,T), measurable with respect to (x,t) for every s ∈ R, and satisfies the following boundedness and coercivity assumptions: 0 <α ≤ a(x,t,s) ≤ β, (2) for almost every (x,t) ∈  × (0,T), for every s ∈ R, where α and β are two positive constants. As far as the data are concerned, we assume that u 0 ∈ L 1 (), E ∈ (L 2 (Q)) N . Mathematics Subject Classification 2000: 35K10, 35K15, 35K65. Key words: Nonlinear parabolic equations, noncoercive problems, infinite energy solutions.