Abstract: In this paper, we have studied the slip-weakening instability on a finite and vertical strike-slip fault with asperity. We assumed that the fault strength is uniform in depth, but the strength is non-uniform in the strike direction, i.e., there is asperity on the fault. The segments with and without asperity of the fault follow the same type of constitutive law, but have different peak stresses. The material surrounding the fault is represented by elastic plates whose upper-surface and bottom are stress-free. We use finite element method to study the evolution of theoretical displacement, stress and strain fields that vary with a remote displacement. When the displacement and frictional stress are not given in advance but are determined in the solving process. According to the results, we have compared the differences of theoretical displacement and the distribution of frictional stress on the fault between stable and unstable slip. In addition, we have compared the results in this paper with known results for the strike-slip fault with a segment of lower strength.