Is the cosmic no hair conjecture true in the Einstein-Maxwell-Dilaton system?

We investigate the gravitational collapsing phenomenon in the spherically symmetric Einstein-Maxwell-dilaton system with a positive cosmological constant. As a preparation, we first study some general properties of the horizons in asymptotically de Sitter spacetime and prove that the area of the: horizons does not decrease and has an upper bound if the matter fields satisfy the dominant energy condition. By using these results, we analytically show that once gravitational collapse occurs from any initial data on a null hypersurface, the system of field equations breaks down inevitably in the domain of outer communications or the boundary, i.e. the black hole event horizon provided that a future null infinity I+ exists, or the asymptotic structure at I+ is broken and the universe will recollapse. In order to clarify which history does the universe trace, we perform a numerical simulation. Then, the dilaton field diverges faster than the logarithmic function almost uniformly and the asymptotic structure would be broken. This implies that the cosmic no hair conjecture is violated in the generalized theory of gravity.