Z. angew. Math. Phys. 48 (1997) 711–724 0044-2275/97/050711-12 $ 1.50+0.20/0 c  1997 Birkh¨auser Verlag, Basel Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP Blow-up vs. global existence for quasilinear parabolic systems with a nonlinear boundary condition Gabriel Acosta 1 and Julio D. Rossi 1,2 Abstract. We study the behavior of positive solutions of the system ut = div(a(u)∇u)+ f (u, v) vt = div(b(v)∇v)+ g(u, v) in Ω a bounded domain with the boundary conditions ∂u ∂η = r(u, v), ∂v ∂η = s(u, v) on ∂Ω and the initial data (u 0 ,v 0 ). We find conditions on the functions a, b, f, g, r, s that guarantee the global existence (or finite time blow-up) of positive solutions for every (u 0 ,v 0 ). Mathematics Subject Classification (1991). 35B35, 35K55, 35B05. Keywords. Parabolic systems, nonlinear boundary conditions, blow up, global existence. I. Introduction Let Ω be a bounded domain in R n with smooth boundary ∂ Ω . In this paper we consider positive solutions of the following system :  u t = div(a(u)∇u)+ f (u,v) in Ω × (0,T ) v t = div(b(v)∇v)+ g(u,v) (1.1) where f (·, ·) and g(·, ·) are positive C 2 functions nondecreasing in each variable and a(·),b(·) are positive (a ≥ c> 0,b ≥ c> 0) , nondecreasing and C 2 . With boundary conditions  ∂u ∂η = r(u,v) on ∂ Ω × (0,T ) ∂v ∂η = s(u,v) (1.2) where r(·, ·),s(·, ·) are positive , nondecreasing in each variable and C 2 . 1 Supported by Universidad de Buenos Aires under grant EX071. 2 Supported by CONICET