Stability in a Semilinear Boundary Value Problem via Invariant Conefields

We give a geometric proof of stability for spatially nonhomogeneous equilibria in the singular perturbation problemut=ε2uxx+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.