  We study a one dimensional metastable dynamics of internal interfaces for the
initial boundary value problem for the following convection-reaction-diffusion
equation \begin{equation*} \partial_t u = \varepsilon \partial_x^2 u
-\partial_x f(u)+ f'(u). \end{equation*} A metastable behavior appears when the
time-dependent solution develops into a layered function in a relatively short
time, and subsequent approaches its steady state in a very long time interval.
A rigorous analysis is used to study such behavior, by means of the
construction of a one-parameter family $\{ U^\varepsilon(x;\xi)\}_\xi$ of
approximate stationary solutions and of a linearization of the original system
around an element of this family. We obtain a system consisting in an ODE for
the parameter $\xi$, describing the position of the interface, coupled with a
PDE for the perturbation $v$, defined as the difference $v:=u-U^\varepsilon$.
The key of our analysis are the spectral properties of the linearized operator
around an element of the family $\{ U^\varepsilon \}$: the presence of a first
eigenvalue, small with respect to $\varepsilon$, leads to a metastable behavior
when $\varepsilon \ll 1$.

Analysis of PDEs