A semilinear problem associated to the space-time fractional heat equation in ${\mathbb{R}}^N$

We study the fully nonlocal semilinear equation ${\mathrm{\partial }}_t^{\alpha }u+(-\mathrm{\Delta }{)}^{\beta }u=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$, $p\ge 1$, where ${\mathrm{\partial }}_t^{\alpha }$ stands for the Caputo derivative of order $\alpha \in (0,1)$ and $(-\mathrm{\Delta }{)}^{\beta }$, $\beta \in (0,1]$, 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. It turns out that every nonnegative nontrivial solution in $L^q$ blows up in finite time if $1<p<p_f:=1+\frac{2\beta }{N}$ whereas if $p\ge p_f$ there are both solutions that blow up and global ones. The critical exponent $p_f$, which does not depend on $\alpha$, coincides with the Fujita exponent for the case $\alpha =1$, in which the time derivative is the standard (local) one. In contrast to the case $\alpha =1$, when $\alpha \in (0,1)$ the critical exponent $p=p_f$ 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.