Reaction–diffusion systems coupled at the boundary and the Morse–Smale property

We study an one-dimensional nonlinear reaction–diffusion system coupled on the boundary. Such system comes from modeling problems of temperature distribution on two bars of same length, jointed together, with different diffusion coefficients. We prove the transversality property of unstable and stable manifolds assuming all equilibrium points are hyperbolic. To this end, we write the system as an equation with noncontinuous diffusion coefficient. We then study the nonincreasing property of the number of zeros of a linearized nonautonomous equation as well as the Sturm–Liouville properties of the solutions of a linear elliptic problem