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

We consider the semilinear diffusion equation $\partial$ t u = Au + |u| $α$ 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|$ξ$| $β$ near the origin $ξ$ = 0, for some $β$ $\in$ (0, 2], then any positive solution of the semilinear diffusion equation blows up in finite time whenever 0 < $α$ $\le$ $β$/(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).

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