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Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws

The aim article is to contribute to the definition of a versatile language for metastability in the context of partial differential equations of evolutive type. A general framework suited for parabolic equations in one dimensional bounded domains is proposed, based on choosing a family of approximate steady states, and on the spectral properties of the linearized operators at such states. The slow motion for solutions belonging to a cylindrical neighborhood of the family of approximate steady states is analyzed by means of a system of an ODE for the parameter $ξ$ that describes the family, coupled with a PDE describing the evolution of the perturbation $v$. We state and prove a general result concerning the reduced system for the couple $(ξ,v)$, called quasi-linearized system, obtained by disregarding the nonlinear term in $v$, and we show how such approach suits to the prototypical example of scalar viscous conservation laws with Dirichlet boundary condition in a bounded one-dimensional interval with convex flux.