We deal with reaction-diffusion equations of bistable type in an inhomogeneous medium. When the reaction term is balanced in the sense that a bulk potential energy attains the same global minimum at the two stable equilibria for each spatial point, we derive a free-boundary problem whose solutions determine equilibirum interfaces. We show that a non-degenerate solution of the free-boundary problem gives rise to an equilibrium internal layer solution of the reaction-diffusion equation, and moreover, the stability property of the latter is obtained from a linearization of the free boundary problem.
"Multi-dimensional stationary internal layers for spatially inhomogeneous reaction-diffusion equations with balanced nonlinearity." Hiroshima Math. J. 33 (3) 391 - 432, November 2003. https://doi.org/10.32917/hmj/1150997983