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

Abstract

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.

Generic hyperbolicity of stationary solutions for a reaction–diffusion system

“…Also, we have that the system is gradient and has global attractor A ⊂ V . For a proof of these facts, see for example [1].…”

“…3) w(0) = z(0) w x (0) + αz x (0) = 0 w(1) = z(1) = 0 (1.4) are different from zero. Since the problem (1.3)-(1.4) defines a closed, self-adjoint and positive operator (see [1] for more details), we have that (u, v) is a hyperbolic equilibrium of (1.1)-(1.2) if zero is not an eigenvalue of the linear system it has been shown in [1] that the reaction-diffusion system (1.1)-(1.2) has a global attractor. Moreover, if all the equilibria are hyperbolic, then the dynamical system generated by the problem is Morse-Smale.…”