A hyperbolic system with relaxation is a system of partial differential equations of hyperbolic type with a zero-order term, describing the relaxation mechanism toward a given equilibrium. After hyperbolic rescaling, the system can be thought as a dynamical system with two time scales: the fast one is governed mainly by the kinetic part of the system itself and it drives the solution toward the equilibrium manifold, the slow one is described by a reduced hyperbolic system of conservation laws.
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Mascia, C. (2008). Stability and Instability Issues for Relaxation Shock Profiles. In: Benzoni-Gavage, S., Serre, D. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75712-2_14
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