Transversality of stable and unstable manifolds for parabolic problems arising in composite materials

In this paper we study one dimensional parabolic problems that arise from composite materials. We show that the eigenvalues and eigenfunctions of the associated linear unbounded operator have the Sturm-Liouville property and the nonincrease of the lap number along the solutions. These results are used to show that the stable and unstable manifolds of equilibrium points are transversal.