Numerical solutions of the nonlinear Vlasov equation show that the long time behavior of an electrostatic plasma wave depends critically on the value of the parameter&ggr;&tgr;(&ggr;is Landau's damping coefficient and&tgr;is the electron bounce time), both for large and small initial wave amplitudes. When the initial wave amplitude is large, the initial nonlinear damping is stronger than Landau damping; for smaller initial amplitudes, the behavior is initially described by Landau's theory. For both types of problems if&ggr;&tgr; > 0.5, the wave damps monotonically; if&ggr;&tgr; < 0.5, the numerical results indicate that the plasma performs amplitude oscillations. When&ggr;&tgr; ≈ 0.5, the plasma displays its critical behavior; here, after some initial damping the wave amplitude levels off at a constant value. The critical behavior observed for mild nonlinear problems is in good agreement with the experimental results of Franklin, Hamberger, and Smith. For a moderately strong nonlinear wave displaying the critical behavior, the wave performs periodic oscillations with a frequency smaller than Landau's linear frequency; this frequency shift is explained because the uniform electron distribution develops a plateau centered at Landau's phase velocity.