Front propagation in infinite cylinders: a variational approach

We present a comprehensive study of front propagation for scalar reaction-diffusion-advection equations in infinite cylinders in the presence of transverse advection by a potential flow and mixtures of Dirichlet and Neumann boundary conditions. We take on a variational point of view, based on the fact that the considered equation is a gradient flow in an exponentially weighted L-space generated by a certain functional, when the dynamics is considered in the reference frame moving with constant velocity along the cylinder axis. In particular, certain traveling wave solutions in the form of fronts connecting different equilibria are critical points of this functional. Under very general assumptions, we prove existence, uniqueness, monotonicity, asymptotic behavior at infinity of the special traveling wave solutions which are minimizers of the considered functional. We also prove that if the functional does not have non-trivial minimizers, there is a traveling wave solution characterized by a certain “minimal speed”. In all cases, the speeds of these waves determine the asymptotic propagation speed of the solutions of the initial-value problem for a large class of initial data that decay sufficiently rapidly exponentially in the direction of propagation. We also perform a detailed variational study of the limit problem arising in the context of combustion theory that leads to a free boundary problem and derive sharp upper and lower bounds for the propagation velocity, as well as establishing convergence of the regularizing approximations to the solution of the free boundary problem. The conclusions of the analysis are illustrated by a number of numerical examples. This study generalizes and extends the existing theory of propagation phenomena in reaction-diffusion equations which is based largely on the applications of the Maximum Principle.