Abstract :
We consider the scalar parabolic equation ut = ϵ2(a2(x)ux)x + ƒ(u), 0 < x < 1, satisfying Neumann boundary conditions and appropriate conditions on the non-linearity ƒ. The attractor Aϵ for the dynamical system generated by this equation has been widely studied in the literature and it is known that, as the diffusion decreases (ϵ → 0), it becomes increasingly complex with an unbounded number of equilibria appearing through bifurcation. Hale and Sakamoto have considered this singular limit in the case of ƒ = ƒ(x, u) and have shown the existence of stable solutions with transition layers when ϵ is sufficiently small and the nonlinearity ƒ satisfies an additional nondegeneracy condition. However, this condition is not satisfied when ƒ = ƒ(u) and the corresponding result do not follow for the problem considered here. We are concerned with the study of the shape and the number of the equilibrium solutions with lower Morse indices, in particular its stable equilibria.
