We consider the semilinear heat equation posed on a smooth bounded domain Ω of R N with Dirichlet or Neumann boundary conditions. The control input is a source term localized in some arbitrary nonempty open subset ω of Ω. The goal of this paper is to prove the uniform large time global nullcontrollability for semilinearities f (s) = ±|s| log α (2 + |s|) where α ∈ [3/2, 2) which is the case left open by Enrique Fernandez-Cara and Enrique Zuazua in 2000. It is worth mentioning that the free solution (without control) can blow-up. First, we establish the small-time global nonnegative-controllability (respectively nonpositive-controllability) of the system, i.e., one can steer any initial data to a nonnegative (respectively nonpositive) state in arbitrary time. In particular, one can act locally thanks to the control term in order to prevent the blow-up from happening. The proof relies on precise observability estimates for the linear heat equation with a bounded potential a(t, x). More precisely, we show that observability holds with a sharp constant of the order exp C a 1/2 ∞ for nonnegative initial data. This inequality comes from a new L 1 Carleman estimate. A Kakutani-Leray-Schauder's fixed point argument enables to go back to the semilinear heat equation. Secondly, the uniform large time null-controllability result comes from three ingredients: the global nonnegative-controllability, a comparison principle between the free solution and the solution to the underlying ordinary differential equation which provides the convergence of the free solution toward 0 in L ∞ (Ω)-norm, and the local null-controllability of the semilinear heat equation. ContentsThe proof of Theorem 1.3 is a consequence of the (global) null-controllability of the linear heat equation with a bounded potential (due to Andrei Fursikov and Oleg Imanuvilov, see [23] or [21, Theorem 1.5]) and the small L ∞ perturbations method (see [3, Lemma 6] and [1], [5], [30], [33], [40] for other results in this direction).The following global null-controllability (positive) result has been proved independently by Enrique Fernandez-Cara, Enrique Zuazua (see [22, Theorem 1.2]) and Viorel Barbu under a sign condition (see [4, Theorem 2] or [6, Theorem 3.6]) for Dirichlet boundary conditions. It has been extended to semilinearities which can depend on the gradient of the state and to Robin boundary conditions (then to Neumann boundary conditions) by