Fujita Critical Exponent

• Fujita critical exponent is the sharp threshold that distinguishes finite-time blow-up from global existence in semilinear evolution equations.
• It extends to diverse operators including fractional, weighted, and sub-Riemannian models by adapting scaling invariance to non-Euclidean contexts.
• Analytic techniques like test-function methods and heat-kernel estimates are used to rigorously identify the critical exponent and its implications.

The Fujita critical exponent is the sharp threshold delineating global existence from finite-time blow-up for solutions to semilinear evolution equations with power-type nonlinearities. In its original context, it quantifies, as a precise function of the geometric and analytic data of the operator, this dichotomy for the classical semilinear heat equation. More broadly, the Fujita exponent admits generalization to a wide spectrum of local, nonlocal, fractional, geometric, weighted, or even forced (inhomogeneous) parabolic PDEs, as well as to discrete and sub-Riemannian analogues. The determination of the Fujita exponent often reveals subtle interplay between scaling, operator geometry, and the nonlinearity, with critical exponents sometimes diverging from those predicted by naive scaling arguments in nonstandard settings (Oza et al., 6 May 2025, Teso et al., 14 Oct 2024, Fino et al., 4 Dec 2025, Chatzakou et al., 6 Nov 2025, Hu et al., 2022).

1. Classical Fujita Exponent: Origin and Scaling Argument

Fujita's classical analysis considered the Cauchy problem for the semilinear heat equation on $\mathbb{R}^n$: $$\partial_t u = \Delta u + u^p,\quad u(x,0) = u_0(x)\geq 0,$$ with $p>1$. By rescaling $$u_\lambda(t,x) = \lambda^\alpha u(\lambda^2 t, \lambda x)$$, invariance requires $\alpha = 2/(p-1)$. The critical exponent emerges by demanding scale-invariance of the $L^1$-norm of the initial mass, yielding the threshold

$p_F(n) = 1 + \frac{2}{n}.$

For $1 < p \leq p_F(n)$, every nontrivial nonnegative solution blows up in finite time; for $p > p_F(n)$, sufficiently small data yield global existence (Needham et al., 11 Nov 2024).

2. Extensions: Geometric, Nonlocal, and Weighted Operators

The paradigm extends to a variety of elliptic, hypoelliptic, and nonlocal diffusion generators. In these settings, the Fujita exponent adapts by replacing the Euclidean dimension $n$ with appropriate geometric, analytic, or effective dimensions. A non-exhaustive table summarizes key representative exponents:

Setting	Effective Dimension	Fujita Exponent ($p_F$)
Euclidean, Laplacian $\Delta$	$n$	$1 + 2/n$
Fractional Laplacian $(-\Delta)^s$, $0<s\leq 1$	$n$	$1 + 2s/n$
Heisenberg group $\mathbb{H}^n$, sub-Laplacian	$Q = 2n+2$	$1 + 2/Q$ or $Q/(Q-2s)$
Hörmander sum-of-squares, subelliptic $L=\sum X_j^2$	$q = \sum \sigma_j$	$1 + 2/q$
Weighted heat: $w(x)=|x_1|^a$ or $|x|^b$	$n+\alpha$	$1 + 2/(n+\alpha)$
Nonlocal w/ weight $a(x)\sim\langle x\rangle^\sigma$	$n$	$1 + (\sigma+2)/n$

In each context, the dichotomy between blow-up and global existence remains anchored at the corresponding $p_F$. For sub-Riemannian models (e.g., Heisenberg group), $Q$ replaces $n$ owing to non-Euclidean volume growth and anisotropic group dilations (Oza et al., 6 May 2025, Georgiev et al., 2019, Chatzakou et al., 6 Nov 2025). Discrete and fractional operators obey analogous criticality, with the order of the diffusion and geometric dimension dictating $p_F$ (Teso et al., 14 Oct 2024).

3. Analytic Techniques for Identifying the Critical Exponent

The methodology for rigorous identification of the Fujita exponent typically involves:

a) Scaling Invariance and Dimensional Analysis:

Equate the scaling exponents of the diffusion operator and the nonlinearity under group or operator-specific dilations. Criticality is achieved when the nonlinearity grows at the same rate as the dissipative term under rescaling, or when the scaling-invariant norm is $L^1$ (Oza et al., 6 May 2025, Teso et al., 14 Oct 2024).

b) Test-Function (Concavity) Methods:

Employ space-time cutoff functions or barrier functions, adapted to the geometry (Euclidean balls, Korányi balls in Heisenberg group), to integrate the weak form and establish contradiction for subcritical exponents, or to construct explicit supersolutions in the supercritical range (Oza et al., 6 May 2025, Chatzakou et al., 6 Nov 2025, Fino et al., 4 Dec 2025).

c) Heat-Kernel and Semigroup Estimates:

Use two-sided Gaussian kernel bounds, semigroup decay in $L^r$ norms, and comparison principles to control the spread of solutions and growth rates. These are essential for both existence and non-existence arguments (Chatzakou et al., 6 Nov 2025).

d) Energy and Maximum Principle Methods:

Leverage weighted $L^p$-energy techniques, interpolation estimates, and maximum principles to propagate norm bounds and preclude global solutions in the subcritical regime (Khomrutai, 2018, Hu et al., 2022).

4. Blow-up/Global-Existence Trichotomy

For a general semilinear evolution equation with superlinear source: $\mathcal{L} u = u^p + f(t,x),\quad u(0,\cdot)=u_0,$ the associated Fujita exponent $p_F$ partitions dynamics as follows (Oza et al., 6 May 2025, Chatzakou et al., 6 Nov 2025):

• Subcritical ($1 < p < p_F$): Every nontrivial nonnegative initial datum leads to finite-time blow-up, regardless of initial size. For nonnegativity, there is no small data threshold.
• Critical ($p = p_F$): Typically, every nontrivial solution blows up in finite time, though special initial profiles with precise decay may admit exceptional (but nonglobal) behavior. For certain weights, geometries, or force terms, logarithmic corrections may appear in the critical dynamics (Chatzakou et al., 6 Nov 2025, Lin et al., 19 Jan 2024).
• Supercritical ($p > p_F$): If the initial data and forcing are sufficiently small (in scaling-critical norms such as $L^1 \cap L^\infty$ or suitably weighted spaces), global-in-time mild (and often classical) solutions exist. For large data, blow-up remains possible.

These regimes are robust across discretizations (Teso et al., 14 Oct 2024), degenerate weights (Hu et al., 2022), and fractional or nonlocal operators (Khomrutai, 2018).

5. Variants: Weighted, Fractional, and Inhomogeneous Problems

A broad range of variants admit explicit, sometimes highly nontrivial, Fujita exponents:

• Fractional and Inhomogeneous Operators: For $(-\Delta)^s$ or more general Lévy-type operators of order $2s$ and spatially varying coefficients, $p_F=1+2s/n$ or suitable modifications arise. Weighted equations with $a(x)\sim\langle x\rangle^\sigma$ yield $p_F=1+(\sigma+2)/n$ (Khomrutai, 2018, Jiang et al., 25 Jul 2024).
• Hörmander and Subelliptic Diffusions: In hypoelliptic frameworks such as spaces with vector fields obeying Hörmander's condition, the effective dimension $q$ defined by the sum of formal weights gives $p_F=1+2/q$ (Chatzakou et al., 6 Nov 2025).
• Space/Time-Dependent Nonlinearities: For weighted nonlinearities, such as $|x|^\alpha |u|^p$ or $t^\eta u^p$, or in equations with time-decaying/explosive forcing, $p_F$ adapts accordingly, for instance as $p_F=1+(\sigma+2(\eta+1))/n$ or more complex expressions in the presence of confining potentials or nonlocal sources (Jiang et al., 25 Jul 2024, Majdoub, 2022).
• Nonlocal Nonlinearities: In fractional heat equations with Riesz potential $I_\alpha(|u|^p)$, the critical exponent depends nontrivially on both diffusion order and convolution: $$p_F(n,\beta,\alpha) = 1 + (\beta+\alpha)/(n-\alpha)$$. This criticality may depart significantly from predictions based solely on scaling, requiring rigorous test-function methods for precise identification (Fino et al., 4 Dec 2025).

6. Discrete, Sublinear, and Non-power Nonlinearities

• Discrete Schemes: For monotone finite difference approximations of reaction-diffusion equations, the discrete Fujita exponent mirrors the continuum theory, with $p_F=1+2s/N$ (local or nonlocal) and convergence of blow-up times under suitable consistency and stability assumptions (Teso et al., 14 Oct 2024).
• Sublinear and Transitional Behavior: In the sublinear regime $0reciprocal transitional exponent $p_c(N)=N/(N+2)$ so that for $p<p_c(N)$, the homogeneous solution is stable, and for $p>p_c(N)$, global dynamics become unstable. This duality is precisely $p_F(N) p_c(N)=1$ (Needham et al., 11 Nov 2024).
• General Nonlinearities: For equations of the form $F(u) = |u|^p \mu(|u|)$, the existence of global solutions at the critical exponent $p_F$ is determined by additional integral conditions on $\mu$, typically $\int_{c_0}^\infty \mu(s)/s\,ds<\infty$. This sharpens the phase portrait at the borderline and extends the classification to weakly non-power, possibly slowly varying nonlinearities (Girardi, 9 Apr 2024).

7. Significance and Further Directions

The critical exponent of Fujita provides a universal, yet highly sensitive, threshold for nonlinearity-dominated blow-up phenomena in semilinear evolution equations. Its determination requires careful structural analysis of both operator and nonlinearity and yields direct operational criteria for blow-up versus global existence across a wide spectrum of models—including parabolic, fractional, kinetic, geometric, weighted, nonlocal, discrete, and sub-Riemannian settings (Oza et al., 6 May 2025, Chatzakou et al., 6 Nov 2025, Teso et al., 14 Oct 2024).

The concept remains central in current research, with ongoing extensions to:

• equations on manifolds and groups with complex geometry or singular weights,
• stochastic and random environments,
• subcritical/critical behavior under perturbations or forced external fields,
• nonlinearities violating standard monotonicity or convexity,
• systems and coupled phenomena,
• sharp lifespan estimates, critical mass blow-up, and singularity formation protocols.

In summary, the Fujita critical exponent constitutes a fundamental organizing principle in the analysis of nonlinear parabolic and related PDEs, encoding the interplay between diffusion, geometry, and growth, and unifying disparate blow-up versus global solvability results under a common scaling-analytic framework (Oza et al., 6 May 2025, Chatzakou et al., 6 Nov 2025, Fino et al., 4 Dec 2025, Teso et al., 14 Oct 2024).