Continuity of the attractors in a singular problem arising in composite materials

In this paper we consider the second-order parabolic problem equation(Pε){utε=(pε(x)uxε)x+c(x)uε+f(uε)in (0,1)∂uε∂n→+b(x)uε=g(uε)for x∈{0,1} where the diffusion coefficient becomes large in a subset which is interior to the interval [0,1][0,1], as ε→0ε→0. We prove the existence of invariant manifolds for (Pε) and using the transversality of invariant manifolds of hyperbolic points obtained in [V.L. Carbone, J.G.Ruas-Filho, Transversality of stable and unstable manifolds for parabolic problems arising in composite materials, J. Math. Anal. Appl. 303 (2005) 220–241] we prove that equation (Pε) and its limit problem are topologically equivalent on the attractors.