We consider the semilinear diffusion equation ∂ t u = Au + |u| α u in the half-space R N + := R N −1 × (0, +∞), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data u(0, x) = u 0 (x) which is nonnegative in the half-space R N + , and the Dirichlet boundary condition u(t, x ′ , 0) = 0 for x ′ ∈ R N −1. We prove that if the symbol of the operator A is of order a|ξ| β near the origin ξ = 0, for some β ∈ (0, 2], then any positive solution of the semilinear diffusion equation blows up in finite time whenever 0 < α ≤ β/(N + 1). On the other hand, we prove existence of positive global solutions of the semilinear diffusion equation in a half-space when α > β/(N + 1). Notice that in the case of the half-space, the exponent β/(N + 1) is smaller than the so-called Fujita exponent β/N in R N. As a consequence we can also solve the blow-up issue for solutions of the above mentioned semilinear diffusion equation in the whole of R N , which are odd in the x N direction (and thus sign changing).