Fujita-Type Critical Exponent

• Fujita-type critical exponent is a threshold criterion that determines whether solutions to nonlinear parabolic equations exhibit finite-time blow-up or global existence.
• It extends the concepts from the classical semilinear heat equation to include nonlocal, degenerate, fractional, and subelliptic operators using scaling and energy methods.
• Insights from this exponent guide numerical discretization strategies and enhance our understanding of long-time dynamics in diverse reaction-diffusion systems.

The Fujita-type critical exponent is a fundamental concept in the theory of nonlinear parabolic and evolution equations, delineating the threshold between universal finite-time blow-up and potential global existence of nontrivial nonnegative solutions. In its classical setting, it is associated with semilinear heat equations, but the notion extends widely to encompass nonlocal, degenerate, fractional, quasilinear, and subelliptic operators in both Euclidean and non-Euclidean (such as the Heisenberg group or Hörmander structures) frameworks. The precise value and nature of the Fujita exponent depend intricately on the underlying operator, spatial geometry, boundary or initial conditions, and the structure of possible lower-order terms or nonlinearities.

1. Origin and Definition of the Fujita Critical Exponent

The archetypal case is the Cauchy problem for the superlinear semilinear heat equation in $\mathbb{R}^N$: $$\partial_t u = \Delta u + u^p, \quad u(x,0) = u_0(x) \geq 0.$$ Fujita (1966) proved that the critical threshold $p_F=1+\frac{2}{N}$ separates two regimes: for $1<p\leq p_F$, any nontrivial nonnegative solution blows up in finite time; for $p>p_F$, global solutions can exist for sufficiently small initial data but large data may still cause blow-up.

This exponent is obtained via scaling analysis and energy methods, comparing the nonlinear growth to the decay rate of the linear heat semigroup. The criticality reflects the balance between the dissipative effect of the diffusion and the amplification by the nonlinearity.

2. Generalizations and Operator-Dependent Exponents

The concept of the Fujita exponent extends to a wide class of operators $L$, including fractional Laplacians, subelliptic (Hörmander) sums of squares, nonlocal and quasilinear diffusions, and to different boundary or initial value problems.

Prototype formulations and exponents:

Equation / Setting	Critical exponent $p_F$	Reference/arXiv ID
$\partial_t u = \Delta u + u^p$ (heat)	$p_F=1+\frac{2}{N}$	(Alfaro, 2016)
Nonlocal: $\partial_t u = J*u-u+u^{1+p}$	$p_F = \beta/N$; $\beta$ from $\hat J(\xi)\sim 1-A|\xi|^\beta$	(Alfaro, 2016)
Fractional: $\partial_t u + (-\Delta)^{s}u = u^p$	$p_F=1+\frac{2s}{N}$	(Bian et al., 2015)
Subelliptic (Hörmander): $\partial_t u - L u = u^p$	$p_F=1+\frac{2}{Q}$, $Q$ = homogeneous dim.	(Chatzakou et al., 6 Nov 2025)
Heisenberg group: Heat/sub-Laplacian	$p_F=1+\frac{2}{Q}$, $Q=2N+2$	(Georgiev et al., 2019, Oza et al., 6 May 2025, Borikhanov et al., 2022)
Weighted/degenerate: $w(x)u_t - \Delta u = u^p$	$p_F=1+\frac{2}{n+\alpha}$, $\alpha=$ weight index	(Hu et al., 2022)
Fractional Rayleigh-Stokes	$p_F=1+\frac{\mu+2(\nu+1)}{N}$ (weights/coefficients)	(Jiang et al., 25 Jul 2024)
Quasilinear nonlocal: $u_t - L_A u \geq (K*u^p)u^q$	$p_F = m-1+\frac{N-\beta+m}{N+\beta}$	(Filippucci et al., 2021)
General linear operator $L$	$p_c=1+\frac{g(n)}{n+n-g(n)}$, $g$ from principal parts	(Girardi, 9 Apr 2024)

For each operator, scaling properties, kernel asymptotics, or spectral geometry determine the effective dimensional parameter appearing in $p_F$.

3. Blow-Up and Dichotomy Theorems

The crux of the Fujita phenomenon is the dichotomy:

• Blow-up: If $1<p\leq p_F$, any nontrivial nonnegative solution experiences finite-time blow-up.
• Global existence: If $p>p_F$, small-data global solutions exist and exhibit decay reflecting the linear semigroup's dissipativity.

Rigorous proofs employ a range of strategies:

• Fourier/Kaplan-type functionals for estimating solution growth (e.g., $f(t) = \int \hat J(\xi)^t \hat u_0(\xi)d\xi$ for nonlocal equations (Alfaro, 2016)).
• Barrier construction via semigroup and supersolution estimates (e.g., $u(t, x)\leq g(t)v(t, x)$ with $v$ solving the linear problem).
• Test-function/energy methods for exterior domains, degenerate or weighted equations, quasilinear or nonlocal sources (Rault, 2010, Hu et al., 2022, Filippucci et al., 2021).

On manifolds or groups with different geometry (Heisenberg, stratified groups), the homogeneous dimension $Q$ replaces the Euclidean $N$ (Chatzakou et al., 6 Nov 2025, Georgiev et al., 2019, Oza et al., 6 May 2025, Borikhanov et al., 2022).

For equations containing combined nonlinearities, such as $u_t-\Delta u = |u|^p + b|\nabla u|^q$, the Fujita exponent surface depends discontinuously on $q$; at $q_c=1+1/(n+1)$ the threshold for $p$ jumps from $1+2/n$ to $\infty$ (Jleli et al., 2019).

4. Influence of Nonlinearity Structure and Forcing Terms

Nonlinearities beyond the pure power-law—such as weighted, space-time-dependent, or nonlocal—modify the critical threshold. Representative cases:

• Weighted reaction: $w(x)u_t - \Delta u = u^p$ lowers $p_F$ to $1+2/(n+\alpha)$ depending on the singularity exponent $\alpha$ (Hu et al., 2022).
• Critical nonlinearity modifier: For $F(u)=|u|^{p_c}\mu(|u|)$, the existence/nonexistence threshold at $p=p_c$ is decided by

$$\int_{s_0}^{\infty} \mu(s) s^{-p_c-1} ds \begin{cases} <\infty & \text{global solutions},\ =\infty & \text{no global solution.} \end{cases}$$

(Girardi, 9 Apr 2024).

• Inhomogeneous/forced cases: For $$u_t + (-\Delta)^d u = |x|^\alpha |u|^p + t^\sigma w(x)$$, the threshold becomes $p_F(\sigma) = (N-2d\sigma+\alpha)/(N-2d\sigma-2d)$ for $-1<\sigma<0$; for $\sigma\geq 0$ it becomes infinite (Majdoub, 2022).
• Riesz-potential-driven or convolution nonlinearity: For $u_t + (-\Delta)^{\beta/2} u = I_\alpha(|u|^p)$, the Fujita exponent is $1+(\beta+\alpha)/(n-\alpha)$, exceeding the scaling exponent $1+(\beta+\alpha)/n$ (Fino et al., 4 Dec 2025).

Such modifications mean that—for nonclassic structures—scaling intuition alone is sometimes insufficient, and capacity, test-function, or detailed analytic techniques are required to locate the actual threshold.

5. Hair-Trigger and Long-Time Dynamics

The critical exponent also determines long-time qualitative behaviors in reaction-diffusion and population dynamics:

• Hair-trigger effect: For Fisher-KPP-type models with nonlocal dispersal, if $p \leq p_F = \beta/N$, any nontrivial initial data lead to invasion, i.e., convergence to the upper state on compacts (even for small perturbations), reflecting the universality of invasion dynamics below the Fujita threshold (Alfaro, 2016).
• Quenching or extinction: For $p>p_F$, small data may be insufficient to sustain global existence, leading to decay to zero over time.

In sublinear regimes $0 < p < 1$ (the sub-Fujita case), sharp algebraic decay rates and a dual "transitional exponent" $p_c(N)=N/(N+2)$ control stability: for $p<p_c(N)$, decay of $u(x, t) - u_h(t)\to 0$; for $p>p_c(N)$, the deviation grows; at $p=p_c(N)$, it remains bounded (Needham et al., 11 Nov 2024).

6. Discrete and Numerical Analogs

Finite-difference, grid-based, or time-stepping approximations of parabolic equations preserve the Fujita-type threshold in the continuum limit. The critical discrete exponent approaches the continuous one, $p_d = 1 + 2/n + O(h^2)$ as grid spacing $h\to 0$. Blow-up times in adaptive time-stepping schemes converge to their continuum counterparts, and the qualitative dichotomy is retained (Teso et al., 14 Oct 2024).

The adaptation of the threshold is required for fractional-difference operators, nonlocal spatial stencils, and other discretization strategies, necessitating careful kernel asymptotic analysis and kernel-dependent scaling.

7. Applications and Ongoing Developments

The Fujita-type exponent remains pivotal in nonlinear evolution PDEs for:

• Classifying blow-up/global existence in broad operator settings
• Designing appropriate numerical discretization/stability criteria
• Understanding the impact of geometry, weights, boundary conditions, or inhomogeneities on solution behavior
• Studying multi-equation and coupled reaction-diffusion systems with interacting nonlinearities (Tréton, 2023, Jiang et al., 25 Jul 2024)
• Connecting the critical threshold to large-time asymptotics and phase transitions in dynamical systems, mathematical biology, or physics

Recent research also identifies subtle discontinuities in the critical surface for combined nonlinearities (Jleli et al., 2019), links the Fujita-Kato transition in damped wave equations to the nature of the damping (Ebert et al., 2020), and extends the threshold to settings where classical scaling predictions are invalid due to nonlocal terms or spatial singularities (Fino et al., 4 Dec 2025).

In summary, the Fujita-type critical exponent is a fundamental, unifying invariant in nonlinear evolution PDEs, providing a sharp dividing line between blow-up and possible small-data global existence, with values and interpretation highly sensitive to operator geometry, kernel decay, nonlinearity form, and system coupling structure.