The issue of the stability of de Sitter space in the context of string theory is indeed a topic of active research and remains an open question. De Sitter space is a maximally symmetric solution to Einstein's equations, which describes an expanding universe with a positive cosmological constant. It is of particular interest in cosmology because it provides a possible explanation for the observed accelerated expansion of our universe.
In string theory, de Sitter space has been studied as a potential model for our universe. However, finding a stable solution that reproduces the properties of our observed universe has proven to be challenging. One of the main difficulties is related to the long-standing problem of constructing a consistent formulation of quantum gravity, which includes string theory.
The discrepancy you mentioned between the evolution of de Sitter space and the evolution of the universe refers to the fact that de Sitter space is a vacuum solution with a constant positive cosmological constant. On the other hand, our universe is not in a pure de Sitter state but contains matter and radiation, which evolve dynamically.
In attempting to incorporate the evolution of the universe, one needs to consider the backreaction of matter and other fields on the geometry of spacetime. This involves solving the coupled equations of motion for both the matter fields and the gravitational field. However, finding a stable solution that accounts for the observed properties of our universe, such as the matter content and the small but nonzero positive cosmological constant, has proven to be a significant challenge.
Several proposals and approaches have been put forward to address the stability issue and to find consistent de Sitter solutions within string theory, such as introducing additional ingredients like branes or fluxes. However, a definitive resolution to this problem is still elusive, and it remains an active area of research in theoretical physics.
It's worth noting that the study of the stability and potential cosmological implications of de Sitter space in the context of string theory is a complex and ongoing research endeavor. New insights and developments may emerge in the future that could shed more light on this topic and potentially provide a satisfactory resolution.