We consider the following fractional reaction-diffusion equation ut (t)+∂t∫0tgα(s)Au(t-s)ds = tγf(u), where gα(t) = tα-1/Γ(α) (0 < α < 1), f ϵ C([0, ∞)) is a non-decreasing function, γ> -1, and A $\mathcal{A}$ is an elliptic operator whose fundamental solution of its associated parabolic equation has Gaussian lower and upper bounds. We characterize the behavior of the functions f so that the above fractional reaction-diffusion equation has a bounded local solution in Lr(ω), for non-negative initial data u0 ϵ Lr(ω), when r > 1 and ω ⊂ ℝN is either a smooth bounded domain or the whole space ℝN. The case r = 1 is also studied.