Summary: The goal is to establish the nonlinear stability of discontinuous steady states, and study the asymptotic behavior of solutions, for the initial- boundary value problem in one space dimension governing incompressible, isothermal shear flow of a non-Newtonian fluid driven by a constant pressure gradient. The fluid is assumed to be highly elastic and viscous; the non-Newtonian contribution to the shear stress satisfies a differential constitutive law characterized by a nonmonotone relation between the total steady shear stress and shear strain-rate that results in steady states having, in general, discontinuities in the strain rate. In a regime where Reynolds number is small compared to Deborah number, it is shown that every solution tends to a steady state as \(t\to \infty\), and steady states that are nonlinearly stable, in a precise sense, are identified.