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

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 $\{ U^\varepsilon(x;ξ)\}_ξ$ 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 $ξ$, 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 $\varepsilon \ll 1$.