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This paper considers the slow motion of the shock layer exhibited by thesolution to the initial-boundary value problem for a scalar hyperbolic systemwith relaxation. Such behavior, known as metastable dynamics, is related to thepresence of a first small eigenvalue for the linearized operator around anequilibrium state; as a consequence, the time-dependent solution approaches itssteady state in an asymptotically exponentially long time interval. In thiscontest, both rigorous and asymptotic approaches are used to analyze such slowmotion for the Jin-Xin system. To describe this dynamics we derive an ODE forthe the position of the internal transition layer, proving how it driftstowards the equilibrium location with a speed rate that is exponentially slow.These analytical results are also validated by numerical computations.

Slow motion of internal shock layers for the Jin-Xin system in one space dimension