Existence and stability of solutions with periodically moving weak internal layers

We consider the periodic parabolic differential equation ε2( ∂2u ∂x2 − ∂u ∂t ) = f (u, x, t, ε) under the assumption that ε is a small positive parameter and that the degenerate equation f (u, x, t, 0) = 0 has two intersecting solutions. We derive conditions such that there exists an asymptotically stable solution up(x, t, ε) which is T -periodic in t, satisfies no-flux boundary conditions and tends to the stable composed root of the degenerate equation as ε→0.