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We show that small homogeneous solutions to the Einstein–Boltzmann-scalar field system exist globally toward the future and tend to the de Sitter solution in a suitable sense. More specifically, we assume that the spacetime is of Bianchi type I–VIII, that the matter is described by Israel particles and that there exists a scalar field with a potential which has a positive lower bound. This represents a generalization of the work [H. Lee and E. Nungesser, Classical Quantum Gravity 35, 025001 (2018)], where a cosmological constant was considered, and a generalization of [H. Lee and J. Lee, J. Math. Phys. 63, 031502 (2022)], where a spatially flat FLRW spacetime was considered. We obtain the global existence and asymptotic behavior of classical solutions to the Einstein–Boltzmann-scalar field system for small initial data.

The Cauchy problem for the Einstein–Boltzmann-scalar field system is studied. A spatially flat Friedmann-Lemaître-Robertson-Walker spacetime is considered with matter contents described by the relativistic Boltzmann equation for Israel particles and a nonlinear scalar field with an exponential potential. The initial data are assumed to be small in a suitable sense, and we obtain the global existence and asymptotic behavior of small solutions.

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