In this paper, we first prove the well-posedness for the nonautonomous reaction-diffusion equations with fractional diffusion in the locally uniform spaces framework. Under very minimal assumptions, then we study the asymptotic behavior of solutions of such equation and show the existence of (H with locally uniform external forces being translation uniform bounded but not translation compact in L p b (R; L q U (R N )). The key to that extensions is a new the space-time estimates in locally

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... rm spaces for the linear fractional power dissipative equation. N 2 Γ(1 − α)). In the limits s → 0 and 2010 Mathematics Subject Classification. Primary: 35K57, 35B40; Secondary: 35B41.