On the Fujita Phenomenon for a Forced Spatio-Temporal Fractional Diffusion Equation
Rihab Ben Belgacem, Mohamed Majdoub
TL;DR
This paper analyzes the Cauchy problem for a semilinear spatio-temporal fractional diffusion equation with forcing: $\partial_t^\alpha u + (-\Delta)^{\mathsf{s}} u = |u|^p + t^\sigma \mathbf{w}(x)$ on $\mathbb{R}^N$, incorporating Caputo time derivatives and nonlocal spatial diffusion. It develops a Duhamel-based framework with kernels $Z$ and $Y$ to capture memory and nonlocal effects, leveraging their self-similar structure and $L^p$ estimates. The main contributions are: (i) local existence of mild solutions and finite-time blow-up in the subcritical regime when the forcing has positive mass, (ii) global existence in the supercritical range for small data and forcing with a forced Fujita-type exponent $p^* = \frac{N\alpha-2\mathsf{s}\sigma}{N\alpha-2\mathsf{s}(\alpha+\sigma)}$, and (iii) two global existence results under distinct smallness/decay assumptions. Collectively, the results illuminate how time–space forcing alters the Fujita threshold in fully nonlocal diffusion and provide criteria for global well-posedness under both global and local smallness conditions.
Abstract
We study the Cauchy problem for a semilinear fractional diffusion equation with a time-dependent forcing term: \[ \partial_t^αu + (-Δ)^{\mathsf{s}} u = |u|^p + t^σ\,\mathbf{w}(x), \quad (t,x) \in (0,\infty) \times \mathbb{R}^N, \] with parameters $α, \mathsf{s} \in (0,1)$, $σ> -α$, and a continuous function $\mathbf{w}$. The operator $\partial_t^α$ denotes the Caputo fractional derivative. Our main contributions are threefold. First, we prove the local existence of mild solutions and demonstrate a finite-time blow-up in the subcritical regime, provided $\int_{\mathbb{R}^N} \mathbf{w}(x)\,dx > 0$. Second, for the supercritical case with $-α< σ< 0$, we establish the global existence for sufficiently small initial data and a forcing term, and we determine the critical exponent to be \[ p_F=\frac{Nα-2\mathsf{s}σ}{Nα-2\mathsf{s}(α+σ)}. \] Finally, for this supercritical range, we prove a more robust global existence result under assumptions requiring only local smallness and controlled growth of the data.