Abstract :
We give a geometric proof of stability for spatially nonhomogeneous equilibria in the singular perturbation problemut= 2uxx+f(x, u),t R+, −1 u 1, with the Neumann boundary conditions onx [0, 1]. The nonlinearity is of the formf(x, u) (1−u2)(u−c(x)), wherec(x) is merely continuous with a finite number of zeros. The strength of the method is in dealing with non-transversal zeros ofc, the case escaping the existing techniques of singular perturbations. The approach is also used for showing existence of unstable equilibria with one transition layer.
