Faults often separate materials with different elastic properties. Nonuniform slip on such faults induces a change in normal stress. That suggests the possibility of self-sustained slip pulses <Weertman, 1980> propagating at the generalized Rayleigh wave speed even with a Coulomb constitutive law (i.e., with a constant coefficient of friction) and a remote driving shear stress that is arbitrarily less than the corresponding frictional strength. Following Andrews and Ben-Zion <1997> (ABZ), we study numerically, with a two-dimensional (2-D) plane strain geometry, the propagation of ruptures along such a dissimilar material interface. However, this problem has been shown to be ill-posed for a wide range of elastic material contrasts <Renardy, 1992; Martins and Sim¿es, 1995; Adams, 1995>. Ranjith and Rice <2000> (RR) showed that when the generalized Rayleigh speed exists, as is the case for the material contrast studied by ABZ, the problem is ill-posed for all values of the coefficient of friction, f, whereas when it does not exist, the problem is ill-posed only for f greater than a critical value. We illustrate the ill-posedness by showing that in the unstable range the numerical solutions do not converge through grid size reduction. By contrast, convergence is achieved in the stable range but, not unexpectedly, only dying pulses are then observed. RR showed that among other regularization procedures, use of an experimentally based law <Prakash and Clifton, 1993; Prakash, 1998>, in which the shear strength in response to an abrupt change in normal stress evolves continuously with time or slip toward the corresponding Coulomb strength, provides a regularization. (Classical slip weakening or rate- and state-dependent constitutive laws having the same kind of abrupt response as Coulomb friction also do not regularize the problem.) Convergence through grid size reduction is then achieved in the otherwise ill-posed range. For sufficiently rapid shear strength evolution, self-sustained pulses are observed. When the generalized Rayleigh wave speed exists, they propagate essentially at that velocity and, in consistence with Weertman's <1980> analysis, the propagation occurs only in one direction, which is that of slip in the more compliant medium. When the generalized Rayleigh wave speed does not exist, similar self-sustained pulses propagate at about the slower S wave speed and in the same direction. RR also suggested that for sufficiently high coefficient of friction, another kind of (less unstable) self-sustained pulses, propagating at a velocity close to the slower P wave speed and in the opposite direction, could also exist. We numerically verify that prediction. ¿ 2001 American Geophysical Union