Asymptotic behavior of interface solutions to semilinear parabolic equations with nonlinear forcing terms

We investigate the asymptotic behavior of solutions to semi linear parabolic equations in bounded intervals. In particular, we are concerned with a special class of solutions, called interface solutions, which exhibit a metastable behavior, meaning that their convergence towards the asymptotic configuration of the system is exponentially slow. The key of our analysis is a linearization around an approximation of the steady state of the problem, and the reduction of the dynamics to a one-dimensional motion, describing the slow convergence of the interfaces towards the equilibrium.