Metastable dynamics of internal interfaces for a convection-reaction-diffusion equation

We study the one-dimensional metastable dynamics of internal interfaces for the initial boundary value problem for the following convection-reaction-diffusion equationpartial derivative(t)u= epsilon partial derivative(2)(x)u - partial derivative(x)f(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 {U-epsilon(x; xi)}(xi) 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 xi, describing the position of the interface coupled with a PDE for the perturbation v and defined as the difference v := u - U-epsilon. The key of our analysis are the spectral properties of the linearised operator around an element of the family {U-epsilon}: the presence of a first eigenvalue, small with respect to epsilon, leads to metastable behaviour when epsilon << 1.