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We present a general procedure to describe slow dynamics in parabolic-hyperbolic systems, under suitable assumptions on the terms appearing in the equations. In particular, our strategy relies in semigroup estimates for the evolution system associated to the family of time-dependent linearized operators around an interfaced configuration. We finally apply the general strategy to the explicit example of an hyperbolic system with relaxation.

Semigroup estimates and fast-slow dynamics in parabolic-hyperbolic systems