We examine axino dark matter in the regime of a low reheating temperature T_R after inflation and taking into account that reheating is a non-instantaneous process. This can have a significant effect on the dark matter abundance, mainly due to entropy production in inflaton decays. We study both thermal and non-thermal production of axinos in the context of the MSSM with ten free parameters. We identify the ranges of the axino mass and the reheating temperature allowed by the LHC and other particle physics data in different models of axino interactions. We confront these limits with cosmological constraints coming the observed dark matter density, large structures formation and big bang nucleosynthesis. We find a number of differences in the phenomenologically acceptable values of the axino mass and the reheating temperature relative to previous studies. In particular, an upper bound on the axino mass becomes dependent on T_R, reaching a maximum value at T_R~10^2 GeV. If the lightest ordinary supersymmetric particle is a wino or a higgsino, we obtain lower a limit of approximately 10 GeV for the reheating temperature. We demonstrate also that entropy production during reheating affects the maximum allowed axino mass and lowest values of the reheating temperature.
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