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: $$\partial_t^\alpha u + (-\Delta)^{\mathsf{s}} u = |u|^p + t^{\sigma}\,\mathbf{w}(x), \quad (t,x) \in (0,\infty) \times \mathbb{R}^N,$$ where $\alpha,\mathsf{s}\in (0,1)$, $\sigma > -\alpha$, and $\mathbf{w}$ is a given continuous function. Here $\partial_t^\alpha$ 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 $$\int\limits_{\mathbb{R}^N} \mathbf{w}(x)\,dx > 0.$$ 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 $$p_F=\frac{N\alpha-2\mathsf{s}\sigma}{N\alpha-2\mathsf{s}(\alpha+\sigma)}.$$ 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.