We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P.~Fife and J.B.~McLeod (1977) without any use of the maximum principle. The method that is illustrated here in the simplest possible setting has been successfully applied to more general parabolic or hyperbolic gradient-like systems.
Thierry Gallay. Emmanuel Risler. "A variational proof of global stability for bistable travelling waves." Differential Integral Equations 20 (8) 901 - 926, 2007. https://doi.org/10.57262/die/1356039363