Abstract
We study the one-dimensional metastable dynamics of internal interfaces for the initial boundary value problem for the following convection-reaction-diffusion equation∂tu=ε∂x2u−∂xf(u)+f′(u). ?>Metastable behaviour appears when the time-dependent solution develops into a layered function in a relatively short time, and subsequently approaches its steady state in a very long time interval. A rigorous analysis is used to study such behaviour by means of the construction of a one-parameter family of approximate stationary solutions and of a linearisation of the original system around an element of this family. We obtain a system consisting of an ODE for the parameter ξ, describing the position of the interface coupled with a PDE for the perturbation v and defined as the difference . The key of our analysis are the spectral properties of the linearised operator around an element of the family : the presence of a first eigenvalue, small with respect to ε, leads to metastable behaviour when .
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