Critical Fujita Exponent in Nonlinear PDEs

• Critical Fujita Exponent is the dimension-dependent threshold in nonlinear parabolic equations that distinguishes between finite-time blow-up and global-in-time solvability.
• Scaling arguments and decay comparisons underpin its derivation, linking the semilinear heat equation’s behavior to critical exponents like p = 1 + 2/n.
• Extensions to weighted, degenerate, and nonlocal contexts illustrate its broad applicability in analyzing geometric, fractional, and anisotropic diffusion phenomena.

The critical Fujita exponent is a dimension-dependent threshold for power-type nonlinear parabolic and related evolutive equations that separates finite-time blow-up (nonexistence of global solutions) from global-in-time small data solvability. It originated in the seminal work of Hiroshi Fujita for the semilinear heat equation and has since been generalized to a broad array of equations, incorporating geometric, weighted, nonlocal, and fractional settings.

1. Classical Statement and Scaling Heuristics

For the standard semilinear heat equation on $\mathbb{R}^n$,

$$u_t = \Delta u + u^p, \qquad u(x,0) = u_0(x) \ge 0,\quad x\in\mathbb{R}^n,$$

the global behavior of solutions with nonnegative, nontrivial data is governed by the competition between diffusive decay and nonlinear growth. The critical (Fujita) exponent is

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

Specifically:

• For $1 < p \le p_F(n)$, every nontrivial nonnegative solution blows up in finite time.
• For $p > p_F(n)$, sufficiently small initial data produce global-in-time solutions, with possible finite-time blow-up only for large data.

This threshold emerges from scaling arguments. Under $$x\mapsto \lambda x,\, t\mapsto \lambda^2 t,\, u\mapsto \lambda^{2/(p-1)} u$$, invariance of the equation occurs precisely if $p=p_F(n)$. The same exponent is obtained by comparing the decay rate of the linear solution in $L^\infty$ ($t^{-n/2}$) with the blow-up rate of the ODE $y' = y^p$ ($t^{-1/(p-1)}$): equating exponents yields $p_F(n)$.

2. Extensions: Potentials, Weights, and Geometry

Recent work generalizes the Fujita exponent to settings with space-dependent coefficients, potentials, anisotropic or degenerate operators, and on non-Euclidean manifolds.

Weighted and potential-perturbed heat equations:

For 1D equations of the form

$$\partial_t u - \partial_x^2 u + V(x)u = (1+x^2)^{-m/2} u^p, \quad x\in \mathbb{R},$$

where $V$ has a ground-state representation $V(x) = \psi''(x)/\psi(x)$ with $\psi(x) \sim |x|^\alpha$ at infinity, and the nonlinearity carries a decaying weight, the critical exponent takes the form (Miyamoto et al., 4 Mar 2025)

$$p_*(\alpha, m) = \begin{cases} 1 + \frac{[2-m]_+}{1+\alpha} & \text{if } \alpha > 1/2, \ \max\bigl\{ \frac{2}{1+2\alpha},\; 1 + \frac{2-m}{1+\alpha} \bigr\} & \text{if } \alpha_* \leq \alpha \leq 1/2, \ \frac{2}{1+2\alpha} & \text{if } -1/2 < \alpha < \alpha_*, \ +\infty & \text{if } \alpha \le -1/2. \end{cases}$$

with a precise dichotomy: blow-up for $p \le p_*(\alpha, m)$; global existence for small data when $p > p_*(\alpha, m)$.

Degenerate diffusion and weighted operators:

For degenerate operators, e.g.,

$$u_t - \text{div}(w(x)\nabla u) = u^p, \quad w(x) = |x_1|^a\ \text{or}\ |x|^b,$$

the critical exponent is $p_*(\alpha) = 1 + 2/(N+\alpha)$, with $\alpha = a$ or $b$, corresponding to the singularity power in the weight (Hu et al., 2022). The weight modifies the effective dimension and thereby the threshold for global existence.

Subelliptic contexts (Heisenberg and Hörmander settings):

On the $n$-dimensional Heisenberg group $\mathbb{H}^n$ with homogeneous dimension $Q = 2n+2$, the Fujita exponent for the heat or damped wave equation is $p_F = 1 + 2/Q$ (Georgiev et al., 2019, Chatzakou et al., 6 Nov 2025, Borikhanov et al., 2022). For general Hörmander systems with homogeneous dimension $q$, the threshold is $p_F = 1 + 2/q$ (Chatzakou et al., 6 Nov 2025).

3. Nonlocal and Fractional Extensions

In nonlocal models, the decay property of the dispersal kernel or the order of the operator determines the critical regime:

Nonlocal diffusion:

$\partial_t u = J * u - u + u^{1+p}$

With $J$'s Fourier symbol $1 - A|\xi|^\beta + o(|\xi|^\beta)$ as $|\xi|\to 0$, the critical Fujita exponent is $p_F = \beta/N$ (Alfaro, 2016).

Fractional Laplacian:

$u_t + (-\Delta)^s u = u^p$

yields $p_F = 1 + 2s/n$ (Euclidean) or $p_F = Q/(Q-2s)$ in the Heisenberg case $(-\Delta_{\mathbb{H}})^s$ (Oza et al., 6 May 2025).

Fractional in time (Caputo derivative):

$\partial_t^\alpha u + L u = |u|^p$

For $0<\alpha<1$ and $L$ a Rayleigh-Stokes or heat-type operator, the exponent remains $p_F = 1 + 2/N$ and is independent of $\alpha$ (Jiang et al., 25 Jul 2024, Kojima, 27 Aug 2024).

4. Fujita-Type Dichotomies in General Evolutions

A unified conclusion for various classes of linear operators $L$:

• For $1 < p \le p_c$, all nontrivial nonnegative solutions (sufficiently regular) blow up in finite time.
• For $p > p_c$, small (in norm) data admit global-in-time, decaying solutions.

Critical exponent calculation:

For general linear PDEs with principal part of $d$-th order in $x \in \mathbb{R}^N$, if the linear $L^1$–$L^\infty$ decay is $\|S(t)\varphi\|_\infty \lesssim t^{-\rho_d}$, and the ODE $y'(t) = y^p$ has blow-up rate $\rho_r=1/(p-1)$, the balance yields

$$\rho_d = \rho_r \implies p_c = 1 + \frac{1}{\rho_d}.$$

Consequences for more general equations and system settings are derived via energy methods, interpolation inequalities, test-function (capacity) arguments, and semigroup decay (Girardi, 9 Apr 2024).

5. Variations and Refinements

Sublinear regime and reciprocals:

For $u_t = \Delta u + u^p$ with $0 < p < 1$ (sublinear), all nontrivial solutions are global, but time-asymptotic stability toward the homogeneous state exhibits a transition at

$p_c(N) = \frac{N}{N+2}$

with the reciprocity $p_F(N) p_c(N) = 1$ (Needham et al., 11 Nov 2024).

Nonlocal nonlinearities and mixed systems:

In systems with combined source and gradient nonlinearities, e.g., $u_t - \Delta u = |u|^p + b|\nabla u|^q$, the critical exponents are piecewise in $p$ and $q$; discontinuities may occur in the global existence regime as parameter boundaries ($q_c=1+1/(n+1)$) are crossed (Jleli et al., 2019). For nonlocal nonlinearities with integral feedback, such as $$u_t = \Delta u + u^\alpha (1 - \sigma \int u^\beta dx)$$, the negative feedback can shift the critical blow-up threshold below the Fujita value (Bian et al., 2015).

Discrete and numerical approximations:

For adaptive-step finite difference schemes approximating reaction-diffusion and nonlocal problems, the discrete Fujita exponent is $p_F^d = 1 + 2s/N$, incorporating the fractional order $s$ of the discretized operator; as grid sizes vanish, the discrete blow-up time converges to the continuous blow-up time (Teso et al., 14 Oct 2024).

6. Proof Techniques and Analytical Structure

The analysis of Fujita-type thresholds centers on:

• Comparison arguments between solutions and suitable supersolutions/subsolutions;
• Energy identities and Gagliardo–Nirenberg inequalities to balance nonlinear growth versus dissipative decay;
• Spectral and semigroup decay analysis for weighted and degenerate models;
• Test-function methods (Kaplan, capacity, or rescaled) using space-time cutoffs to reach contradictions in subcritical regimes;
• Fixed-point/iteration in scaling-critical functional settings for global solvability proofs in the supercritical case;
• Refined integral criteria (e.g., imposing $\int_0^\varepsilon \mu(s) s^{-1-p_c} ds$ on nonlinearities $|u|^p \mu(|u|)$) to distinguish exact thresholds in generalized models (Girardi, 9 Apr 2024).

7. Open Problems and Future Directions

• Critical and borderline cases: Precise blow-up mechanisms and sharp lifespan estimates at the threshold $p=p_c$, especially in anisotropic, degenerate, or higher-dimensional settings.
• Systems and mixed nonlinearities: Extensions to systems, equations involving more general nonlinear dependencies (e.g., with gradients or fractional temporal dynamics), and inhomogeneous operators.
• Geometric and non-Euclidean diffusion: Comprehensive understanding for stratified groups, variable coefficients, and evolutionary equations on manifolds or metric measure spaces.
• Numerical phenomena: Rigorous convergence of discrete thresholds, rates, and blow-up profiles in sophisticated time-adaptive and space-fractional settings.

The theory of the critical Fujita exponent continues to play a central role in nonlinear evolution PDEs by quantifying the delicate balance between diffusive dispersion and nonlinear amplification, with ongoing advances in both analytical and geometric contexts.