We consider the family of semilinear parabolic problemswhere a > 0, Ω is the unit square, Ω = h (Ω), h is a family of diffeomorphisms which converge to the identity of Ω in C 0,α -norm, 0 ≤ α < 1, but not in the C 1 -norm and, f, g : R → R are real functions.Under appropriate hypothesis, we show that the limiting problem is given bywhere µ is essentially the limit of the jacobian determinant of the diffeomorphism h : ∂Ω → ∂h (Ω).We prove that the problem is well posed for 0 ≤ ≤ 0 , 0 > 0, in a suitable phase space, the associated semigroup has a global attractor A and the family {A } 0 ≤ ≤ 0 is continuous at = 0.