Abstract: We study the Einstein-scalar field system with positive cosmological constant and spherically symmetric characteristic initial data. We prove well-posedness, global existence and exponential decay in (Bondi) time, for small data. From this, it follows that initial data close enough to de Sitter data evolves, according to the system under consideration, to a causally geodesically complete space-time which approaches de Sitter asymptotically at an exponential rate; this is a non-linear stability result for de Sitter within the class under consideration as well as a realization of the cosmic \"no hair\" conjecture.
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