A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type

We identified a variational structure associated with traveling waves for systems of reaction-diffusion equations of gradient type with equal diffusion coefficients defined inside an infinite cylinder, with either Neumann or Dirichlet boundary conditions. We show that the traveling wave solutions that decay sufficiently rapidly exponentially at one end of the cylinder are critical points of certain functionals. We obtain a global upper bound on the speed of these solutions. We also show that for a wide class of solutions of the initial value problem an appropriately defined instantaneous propagation speed approaches a limit at long times. Furthermore, under certain assumptions on the shape of the solution, there exists a reference frame in which the solution of the initial value problem converges to the traveling wave solution with this speed at least on a sequence of times. In addition, for a class of solutions we establish bounds on the shape of the solution in the reference frame associated with its leading edge and determine accessible limiting traveling wave solutions. For this class of solutions we find the upper and lower bounds for the speed of the leading edge.