We perform numerical computations to investigate the dynamical evolution of axisymmetric gravitational waves in asymptotically de Sitter spacetime. The final fates of such spacetimes are classified into two types: (1) de Sitter spacetime with small perturbations, and (2) Schwarzschild de Sitter like spacetime with both black-hole and cosmological apparent horizons. We also find that if the mass of gravitational waves is larger than the critical value, Mcrit(3) -1, then the black hole is not formed even in the case that there exist highly nonlinear localized gravitational waves. In such a case, the Universe is initially dominated by gravitational waves rather than a cosmological constant and eventually approaches a de Sitter universe. These results are consistent with the previous analysis by use of initial data.
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