Linda Maria De Cave[a] and Marta Strani[b]
Asymptotic behavior of interface solutions to semilinear parabolic equations with nonlinear forcing terms
Received: 12 March 2018
Accepted in revised form: 7 August 2018
Mathematics Subject Classification (2010): 35B25, 35B36, 35B40, 35K45.
Keywords: Metastability, slow motion, internal interfaces, asymptotic dynamics, semilinear diffusion.
[a]: Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland.
[b]: Università Ca' Foscari, Dipartimento di Scienze Molecolari e Nanosistemi, Via Torino 155, 30172, Venezia Mestre, Italy.
Abstract: We investigate the asymptotic behavior of solutions to semilinear 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.