Metastability for Nonlinear Parabolic Equations with Application to Scalar Viscous Conservation Laws
Corrado Mascia, Marta Strani · SIAM Journal on Mathematical Analysis · 2013
The aim of this paper 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 $\{U^{\varepsilon}(\cdot;\xi) \}_{{}_{\xi \in J}}$ 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 $\{U^{\varepsilon}\}$ is analyzed by means of a system of an ODE for the parameter $\xi=\xi(t)$, coupled with a PDE describing the evolution of the perturbation $v:=u-U^\varepsilon(\cdot;\xi)$. We state and prove a general result concerning the reduced system for the couple $(\xi,v)$, called quasi-linearized system, obtained by disregarding the nonlinear term in $v$, and we show how such an approach suits to the prototypical example of scalar viscous conservation laws with Dirichlet boundary conditions in a bounded one-dimensional interval with convex flux.