The nonlinear evolution of the resonant absorption of standing and propagating Alfv¿n waves in an inhomogeneous plasma is studied via solution of the time-dependent, three-dimensional, low-&bgr;, resistive MHD equations over a wide parameter range. When the nonlinear effects become important, the velocities at the dissipation layer were found to be lower than the linear scaling of S1/3 would predict, where S is the Lundquist number. Highly sheared velocities that are subject to the Kelvin-Helmholtz-like instability were found at the narrow dissipation layers. Three-dimensional Kelvin-Helmholtz-like vortices appear at and near the dissipation layers and propagate along the slab of plasma when traveling Alfv¿n wave solution are considered. The narrow resonant heating layers are deformed by the self-consistent shear flow. In the solar active regions where the resonant absorption of Alfv¿n waves is believed to occur, the instability may lead to turbulent enhancement of the dissipation parameters and account for the observed turbulent velocities inferred from the nonthermal broadening of x-ray and EUV emission lines. The self consistent J¿B force changes significantly the density structure of the loop that leads to a shift in the global mode frequency response of the loop and a subsequent drop in the heating rate. In the solar corona the density evolution of the loop is likely to be dominated by evaporation of material from the transition region. ¿ American Geophysical Union 1995