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

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

Problem Statement
This paper investigates the Cauchy problem for a semilinear spatio-temporal fractional diffusion equation subject to a time-dependent external forcing term. The equation is given by:
$$\partial_t^\alpha u + (-\Delta)^s u = |u|^p + t^\sigma w(x), \quad (t, x) \in (0, \infty) \times \mathbb{R}^N,$$
with initial condition $u(0, x) = u_0(x) \geq 0$. Here, $\partial_t^\alpha$ denotes the Caputo fractional derivative of order $\alpha \in (0, 1)$, $(-\Delta)^s$ is the fractional Laplacian of order $s \in (0, 1)$, $p > 1$ is the nonlinearity exponent, and $\sigma > -\alpha$ characterizes the temporal growth or decay of the forcing term $w(x)$. The primary objective is to characterize the "Fujita phenomenon"—specifically, to identify the critical exponent $p_F$ that separates the regime of finite-time blow-up from the regime of global existence for solutions.

Methodology
The authors employ a combination of functional analysis, fractional calculus, and scaling arguments to analyze the problem:

1. Mild Solution Framework: The study utilizes the Duhamel representation formula, expressing the solution $u$ as the sum of three components: the linear evolution of the initial data ($Lu_0$), the nonlinear reaction term ($Nu$), and the forced term ($Sw$). This relies on the fundamental solution kernels $Z(t, x)$ and $Y(t, x)$ associated with the linear operator $\partial_t^\alpha + (-\Delta)^s$.
2. Kernel Estimates: A rigorous analysis of the scaling properties, integrability, and regularity of the kernels $Z$ and $Y$ is conducted. These estimates, derived from the self-similar structure of the fundamental solutions, are essential for establishing contraction mappings in appropriate Lebesgue spaces.
3. Fixed Point Arguments: Local and global existence results are proven using the Banach fixed-point theorem in suitable function spaces (specifically $L^\infty(0, T; L^q(\mathbb{R}^N))$ and weighted spaces involving time decay).
4. Test Function Method (Non-existence): To prove blow-up results, the authors adapt the test function method. They construct a carefully rescaled test function $\psi_R(t, x)$ and apply fractional integration-by-parts formulas (involving the right-sided Riemann-Liouville derivative) to derive integral inequalities. By analyzing the asymptotic behavior of these inequalities as the scaling parameter $R \to \infty$, they derive contradictions for subcritical exponents.
5. Local Average Criteria: For global existence, the paper introduces a criterion based on the local averages of the initial data and forcing term over balls, rather than relying solely on global norm smallness.

Key Contributions and Results

• Local Well-Posedness: The paper establishes the local-in-time existence and uniqueness of mild solutions in $L^q(\mathbb{R}^N)$ for initial data $u_0$ and forcing $w$ in $L^q$, provided $q$ is within a specific range relative to the critical exponent.
• Finite-Time Blow-Up (Subcritical Regime): Under the assumption that the forcing term has positive total mass ($\int_{\mathbb{R}^N} w(x) dx > 0$), the authors prove that no global weak solutions exist if the exponent $p$ falls below a specific critical threshold.
  • For $-\alpha < \sigma \leq 0$ and $2s < N$, the critical exponent is identified as:
    $$p^* = \frac{N\alpha - 2s\sigma}{N\alpha - 2s(\alpha + \sigma)}.$$
    If $1 < p < p^*$, solutions blow up in finite time.
  • If $\sigma > 0$, blow-up occurs for all $p > 1$.
• Global Existence (Supercritical Regime):
  • Small Data Result: For $p \geq p^*$, global solutions are proven to exist if the initial data $u_0$ and the forcing term $w$ are sufficiently small in their respective critical norms ($L^{p_c}$ and $L^{r_c}$).
  • Robust Existence Criterion: A more general global existence result is established for the supercritical range. This result relaxes the requirement for global smallness, instead requiring only that the local averages of $u_0$ and $w$ over large balls satisfy specific decay and smallness conditions. This allows for data that may be large in global norms but decays appropriately at infinity.
• Critical Exponent Identification: The paper explicitly derives the sharp Fujita-type threshold $p^*$ for the fully fractional equation with time-growing forcing. The authors note that this threshold differs from the unforced case ($p_F = 1 + \frac{2s\alpha}{N\alpha + 2s(1-\alpha)}$), demonstrating that the external forcing accelerates the blow-up mechanism by effectively lowering the critical exponent.

Significance and Claims
The paper claims to provide the first sharp Fujita-type threshold for fully spatio-temporal fractional diffusion equations with time-growing external forcing. The significance lies in:

1. Generalization: Extending the classical Fujita phenomenon (originally for the heat equation) to models combining memory effects (fractional time) and long-range spatial interactions (fractional Laplacian) with external sources.
2. Forcing Impact: Quantifying how the temporal power law of the forcing term ($t^\sigma$) interacts with the fractional diffusion parameters ($\alpha, s$) to alter the critical exponent. The results indicate that forcing terms can counterbalance the regularizing effects of fractional diffusion, expanding the range of exponents leading to blow-up.
3. Refined Existence Criteria: Moving beyond standard smallness assumptions in global norms to provide existence criteria based on the spatial distribution and local averages of the data, offering a more nuanced understanding of global solvability.

The authors conclude that their analytical findings are supported by numerical simulations in one dimension, which visually confirm the predicted transition between blow-up and bounded behavior at the derived critical exponent.