We study the fully nonlocal semilinear equation \(\partial _t^\alpha u+(-\Delta )^\beta u=|u|^{p-1}u\) , \(p\geqslant 1\) , where \(\partial _t^\alpha \) stands for the usual time derivative when \(\alpha =1\) and for the Caputo \(\alpha \) -derivative if \(\alpha \in (0,1)\) , while \((-\Delta )^\beta \) , \(\beta \in (0,1]\) , is the usual \(\beta \) power of the Laplacian. We prescribe an initial datum in \(L^q(\mathbb {R}^N)\) . We give conditions ensuring the existence and uniqueness of a solution living in \(L^q({\mathbb {R}}^N)\) up to a maximal existence time T that may be finite or infinite. If T is finite, the \(L^q\) norm of the solution becomes unbounded as time approaches T, and u is said to blow up in \(L^q\) . 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. Our weakest condition for global existence and our condition for blow-up are both related to the size of the averages of the initial datum in balls. As a corollary, every nonnegative nontrivial solution in \(L^q\) blows up in finite time if \(1<p<p_f:=1+\frac{2\beta }{N}\) whereas if \(p> p_f\) there are both solutions that blow up and global ones. Noteworthy, the critical Fujita-type exponent \(p_f\) does not depend on \(\alpha \) . However, there is an important difference in the behavior of solutions in the critical case \(p=p_f\) depending on the value of this parameter: when \(\alpha =1\) it was known that all nonnegative and nontrivial solutions blow up, while we prove here that if \(\alpha \in (0,1)\) there is global existence for some initial data.
