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We consider the semilinear diffusion equation $\partial$ t u = Au + |u|$\alpha$ u in the half-space R N + := R N --1 x (0, +$\infty$), where A is alinear diffusion operator, which may be the classical Laplace operator, or afractional Laplace operator, or an appropriate non regularizing nonlocaloperator. 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 boundarycondition u(t, x ' , 0) = 0 for x ' $\in$ R N --1. We prove that if the symbolof the operator A is of order a|$\xi$| $\beta$ near the origin $\xi$ = 0, forsome $\beta$ $\in$ (0, 2], then any positive solution of the semilineardiffusion equation blows up in finite time whenever 0 < $\alpha$ $\le$$\beta$/(N + 1). On the other hand, we prove existence of positive globalsolutions of the semilinear diffusion equation in a half-space when $\alpha$ >$\beta$/(N + 1). Notice that in the case of the half-space, the exponent$\beta$/(N + 1) is smaller than the so-called Fujita exponent $\beta$/N in R N.As a consequence we can also solve the blow-up issue for solutions of the abovementioned semilinear diffusion equation in the whole of R N , which are odd inthe x N direction (and thus sign changing).

Blow-up phenomena for positive solutions of semilinear diffusion equations in a half-space: the influence of the dispersion kernel