On the Fujita Phenomenon for a Forced Spatio-Temporal Fractional Diffusion Equation

We investigate the Cauchy problem for a semilinear spatio–temporal fractional diffusion equation with a time-dependent forcing term: $${\mathrm{\partial }}_t^{\alpha }u+(-\mathrm{\Delta }{)}^{\mathsf{s}}u=\mathrm{\mid }u{\mathrm{\mid }}^p+t^{\sigma }\mathbf{w}(x),(t,x)\in (0,\mathrm{\infty })\times {\mathbb{R}}^N,$$ where $\alpha ,\mathsf{s}\in (0,1)$, $\sigma >-\alpha$, and $\mathbf{w}$ is a given continuous function. Here ${\mathrm{\partial }}_t^{\alpha }$ denotes the Caputo fractional derivative. Our main results are threefold. First, we establish local-in-time existence of mild solutions and prove finite-time blow-up in the subcritical regime, under the positivity condition $$\underset{{\mathbb{R}}^N}{\int }\mathbf{w}(x)dx>0.$$ Second, in the supercritical case $-\alpha <\sigma <0$, we prove the global existence of solutions for sufficiently small initial data and forcing term, and we identify the corresponding critical exponent as $$p_F=\frac{N\alpha -2\mathsf{s}\sigma }{N\alpha -2\mathsf{s}(\alpha +\sigma )}\mathrm{.}$$ Finally, within this supercritical range, we obtain a more robust global existence result under weaker assumptions that require only local smallness and controlled growth of the data. To the best of our knowledge, a sharp Fujita-type threshold for fully spatio-temporal fractional diffusion equations with time-growing external forcing has not been previously established.