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This paper considers the slow motion of the shock layer exhibited by the solution to the initial-boundary value problem for a scalar hyperbolic system with relaxation. Such behavior, known as , is related to the presence of a first small eigenvalue for the linearized operator around an equilibrium state; as a consequence, the time-dependent solution approaches its steady state in an asymptotically exponentially long time interval. In this contest, both rigorous and asymptotic approaches are used to analyze the slow motion of solutions to the Jin–Xin system. To describe this dynamics, we derive an ODE for the position of the internal transition layer, proving that it drifts towards 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