A semilinear problem associated to the space-time fractional heat equation in RN

We study the fully nonlocal semilinear equation ₜ^ u+ (-) ^ u=|u|^p-1u, p1, where ₜ^ stands for the Caputo derivative of order (0, 1) and (-) ^, (0, 1], is the usual power of the Laplacian. We prescribe an initial datum in Lq (RN). We give conditions ensuring the existence and uniqueness of a solution living in Lq (RN) up to a maximal existence time T that may be finite or infinite. If~T is finite, the Lq norm of the solution becomes unbounded as time approaches T, and u is said to blow up in Lq. Otherwise, the solution is global in time. For the case of nonnegative and nontrivial solutions, we give conditions on the initial datum that ensure either blow-up or global existence. It turns out that every nonnegative nontrivial solution in Lq blows up in finite time if 1<p<pf: =1+2 whereas if p pf there are both solutions that blow up and global ones. The critical exponent pf, which does not depend on, coincides with the Fujita exponent for the case =1, in which the time derivative is the standard (local) one. In contrast to the case =1, when (0, 1) the critical exponent p=pf falls within the situation in which global existence may occur. Our weakest condition for global existence and our condition for blow-up are both related to the size of the mean value of the initial datum in large balls.