  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 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 ' $\in$ R N --1. We prove that if the symbol
of the operator A is of order a|$\xi$| $\beta$ near the origin $\xi$ = 0, for
some $\beta$ $\in$ (0, 2], then any positive solution of the semilinear
diffusion equation blows up in finite time whenever 0 < $\alpha$ $\le$
$\beta$/(N + 1). On the other hand, we prove existence of positive global
solutions 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 above
mentioned semilinear diffusion equation in the whole of R N , which are odd in
the x N direction (and thus sign changing).
