Instability of hysteretic phase interfaces in a mean-field model with inhomogeneities

We study a system of non-identical bistable particles that is driven by a dynamical constraint and coupled through a non-local mean-field. Assuming piecewise affine constitutive laws we prove the existence of traveling wave solutions and characterize their dynamical stability. Our findings explain the two dynamical regimes for phase interface that can be observed in numerical simulations with different parameters. We further discuss the convergence to a rate-independent model with strong hysteresis in the limit of vanishing relaxation time.