An. S ¸t. Univ. Ovidius Constant ¸a Vol. 19(1), 2011, 121–138 EXISTENCE AND ASYMPTOTIC BEHAVIOUR OF POSITIVE SOLUTIONS FOR SOME NONLINEAR PARABOLIC SYSTEMS IN THE HALF-SPACE Abdeljabbar Ghanmi and Faten Toumi Abstract We are concerned with the nonlinear parabolic system Δu - au - ∂u ∂t = λp(x, t)f (v) , Δv - bv - ∂v ∂t = μq(x, t)g (u) , in R n + × (0, ∞), subject to some Dirichlet boundary conditions, where the potentials p, q, a and b are allowed to satisfy some hypotheses related to the parabolic Kato class P ∞ (R n + ), the functions f and g are nonneg- ative nondecreasing and continuous. More precisely, we shall prove the existence of positive continuous solutions with precise global behaviour. We will use some potential theory arguments. 1 Introduction In this work, we deal with the existence of positive continuous solutions ( in the sense of distributions) and their asymptotic behaviour for the following Key Words: Green function, Parabolic systems, Positive solution. Mathematics Subject Classification: 34B27, 35K10 Received: April, 2010 Accepted: December, 2010 121