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.