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

We study the Cauchy problem for a semilinear spatio-temporal fractional diffusion equation on the whole Euclidean space. The model involves a Caputo time derivative of order 0<α<1, a fractional power of the Laplacian in space, and a source term consisting of a power-type nonlinearity together with a time-dependent forcing term of the form tσw(x), where w is a continuous spatial weight. The temporal exponent σ is allowed to satisfy σ>-α, thereby covering both decaying and growing forcing effects. Our main contributions are threefold. First, we prove local-in-time existence of mild solutions and establish finite-time blow-up in the subcritical regime, assuming that the spatial weight has positive integral over the whole space. Second, in the supercritical case associated with negative temporal exponents σ∈(-α,0), we show global existence for sufficiently small initial data and forcing, and derive the corresponding Fujita-type critical exponent explicitly in terms of the spatial dimension, the temporal and spatial fractional orders, and the growth parameter σ. Third, within the same supercritical range, we obtain a stronger global existence result under weaker hypotheses requiring only local smallness and suitable growth control of the data. To the best of our knowledge, this is the first work to establish a sharp Fujita-type threshold for fully spatio-temporal fractional diffusion equations with time-growing external forcing.