--- title: Fujita Exponent in Semilinear Heat Equations url: https://www.emergentmind.com/topics/fujita-exponent type: topic --- # Fujita Exponent in Semilinear Heat Equations Searching arXiv for recent and foundational papers on the Fujita exponent to ground the article in published work. The **Fujita exponent** is the critical power that separates universal finite-time blow-up from the possibility of global-in-time solutions for semilinear heat-type equations with nonnegative data. In the classical Cauchy problem on \(\mathbb{R}^N\), \[ u_t-\Delta u=u^p,\qquad u(x,0)=u_0(x)\ge 0, \] Fujita discovered that the critical value is \[ p_F=1+\frac{2}{N}, \] with the following dichotomy: if \(1p_F\), sufficiently small initial data generate global-in-time positive solutions [1510.07832]. Subsequent work has shown that this phenomenon persists across weighted, fractional, nonlocal, subelliptic, damped, and geometric settings, but with the critical exponent modified by the effective diffusion order, forcing, homogeneous dimension, or nonlocal structure [2212.12491], [2408.01577], [2211.10759], [1908.02989], [2512.04506]. ## 1. Classical threshold and its meaning In the classical semilinear heat equation, \[ u_t-\Delta u=u^p \quad \text{on } \mathbb{R}^N\times(0,\infty), \] the Fujita exponent is the borderline between a reaction-dominated regime and a diffusion-dominated regime [1510.07832]. For \(1p_F\), small initial data may produce global solutions, although sufficiently large data may still blow up [1510.07832]. The same threshold is described in several equivalent notational conventions across the literature. One convention writes the exponent as \(p_F=1+\frac{2}{N}\) [1510.07832], while other works parameterize the same threshold in terms of the diffusion decay or homogeneous dimension and obtain formulas such as \(1+\frac{n}{2}\), \(1+\frac{\alpha}{n}\), or \(1+\frac{2s}{N}\), depending on the operator and normalization [2212.12491], [1706.01251], [2408.01577]. This suggests that the numerical form of the critical exponent is model-dependent, but its function is structurally stable: it marks the transition between unavoidable blow-up and small-data global existence. A persistent theme is that the Fujita exponent is tied to the large-time behavior of the linear semigroup. In the classical case, the heat kernel decays like \(t^{-N/2}\), and this decay determines the balance with the power nonlinearity. Several of the cited works make this principle explicit by replacing the Euclidean dimension \(N\) with an effective dimension, a homogeneous dimension, or a fractional diffusion order [2212.12491], [1908.02989], [2511.04196]. ## 2. Geometric and operator-dependent generalizations A major direction in modern Fujita theory is the replacement of the Laplacian by non-Euclidean or degenerate diffusion operators. For the weighted semilinear heat equation \[ u_t-w(x)^{-1}\operatorname{div}(w(x)\nabla u)=u^p, \] with weights \(w(x)=|x_1|^a\), \(a\in[0,1)\), or \(w(x)=|x|^b\), \(b\in[0,n)\), the critical Fujita exponent is \[ p^*(\alpha)=1+\frac{n+\alpha}{2},\qquad \alpha\in\{a,b\}, \] and the measure growth satisfies \(w(B(x,r))\simeq r^{n+\alpha}\) [2212.12491]. In this setting, the effective dimension is \(n+\alpha\), reflecting the degeneracy of the weighted diffusion. If \(1p^*(\alpha)\), sufficiently small data in suitable weighted Lorentz spaces yield unique global solutions [2212.12491]. On stratified Lie groups, the homogeneous dimension \(Q\) replaces the Euclidean dimension. For the semilinear heat equation with forcing \[ u_t-\mathcal{L}_G u=|u|^p+f(x) \] on a stratified Lie group \(G\), the Fujita exponent is \[ p_F=\frac{Q}{Q-2}, \] and for \(Q\in\{1,2\}\) one has \(p_F=+\infty\) [2211.10759]. The same homogeneous-dimension mechanism appears on the Heisenberg group. For the semilinear heat equation with forcing on \(\mathbb{H}^N\), \[ u_t-\Delta_H u=|u|^p+f(\eta), \] the critical exponent is \[ p_s=\frac{Q}{Q-2},\qquad Q=2N+2, \] which is finite for all \(N\ge1\), in contrast to the Euclidean forced case in dimensions \(1\) and \(2\) [2207.03744]. For the semilinear damped wave equation on the Heisenberg group, \[ u_{tt}-\Delta_{\mathbb{H}}u+u_t=|u|^p, \] the critical exponent becomes \[ p_{\mathrm{Fuj}(\mathscr{Q})}=1+\frac{2}{\mathscr{Q}},\qquad \mathscr{Q}=2n+2, \] showing that strong damping induces a parabolic Fujita-type threshold governed by the homogeneous dimension [1908.02989]. A further extension replaces \(\Delta\) by sums of squares of Hörmander vector fields. For \[ u_t-\sum_{j=1}^m X_j^2u=u^p, \] the critical exponent is \[ p_F=1+\frac{2}{q}, \] where \(q\) is the homogeneous dimension associated with the anisotropic dilations of the Hörmander system [2511.04196]. In this framework, the Carnot–Carathéodory geometry, volume growth \(|B_X(x,r)|\sim r^q\), and Gaussian bounds for the heat kernel in the CC metric determine the threshold [2511.04196]. ## 3. Fractional, mixed local–nonlocal, and subelliptic diffusions For fractional diffusion, the order of the operator modifies the Fujita exponent. For the space-fractional equation \[ (\partial_t+(-\Delta)^{\alpha/2})u=u^p, \] the Fujita exponent is \[ p_F=1+\frac{\alpha}{n}, \] and if \(11+\frac{2s}{N}\), there exist global solutions for some nonnegative initial data [2408.01577]. The linear asymptotics are fractional: the mixed kernel behaves at large times like the fractional heat kernel, and this governs the critical exponent [2408.01577]. A related sub-Riemannian generalization appears on the Heisenberg group with fractional sub-Laplacian and forcing, \[ \partial_t u+(-\Delta_{\mathbb H^N})^s u=|u|^p+f, \] where the critical exponent is \[ p_F=\frac{Q}{Q-2s},\qquad Q=2N+2, \] with global existence in the supercritical case, nonexistence in the subcritical case, and finite-time blow-up in the critical case for a class of forcing terms [2505.03619]. The homogeneous dimension \(Q\) and the diffusion order \(2s\) play the same role here that \(N\) and \(2\) play in the Euclidean heat equation [2505.03619]. ## 4. Nonlocal reaction structures and departures from classical scaling Not all Fujita exponents are determined by naive scaling. For the fractional heat equation with Riesz-potential nonlinearity \[ u_t+(-\Delta)^{\beta/2}u=I_\alpha(|u|^p), \] the paper introduces the Fujita-type critical exponent \[ p_{\mathrm{Fuj}(n,\beta,\alpha)}=1+\frac{\beta+\alpha}{n-\alpha}, \] while the scaling-based exponent is \[ p_{sc}=1+\frac{\beta+\alpha}{n}. \] The crucial point is that \[ p_{\mathrm{Fuj}(n,\beta,\alpha)}>p_{sc} \] for \(\alpha>0\), so the actual threshold is not governed by scaling [2512.04506]. The paper proves finite-time blow-up for \[ \frac{n}{n-\alpha}p_{\mathrm{Fuj}(n,\beta,\alpha)}\) [2512.04506]. This is explicitly compared with earlier results of Cazenave et al. for a heat equation with time-nonlocal nonlinearity, where the critical exponent is likewise not given by the usual scaling argument [2512.04506]. Nonlocal diffusion kernels can also alter the Fujita exponent through their tails. For \[ \partial_tu=J*u-u+u^{1+p}, \] the decisive quantity is the behavior of \(\widehat{J}(\xi)\) near \(\xi=0\), \[ \widehat{J}(\xi)=1-A|\xi|^\beta+o(|\xi|^\beta), \] with \(0<\beta\le2\) [1605.00891]. If \(J\) has compact support, exponential decay, or finite second moment, then \(\beta=2\) and the Fujita exponent is of heat type; for algebraic tails one may obtain a fractional-type threshold depending on whether the second moment is finite [1605.00891]. This suggests that the Fujita exponent is determined not only by the formal operator but also by the dispersal tail encoded in the linear kernel. A different nonlocal modification occurs in the equation \[ u_t-\Delta u=u^\alpha\Bigl(1-\sigma\int_{\mathbb{R}^n}u^\beta\,dx\Bigr). \] For \(\beta=1\) and \(n\ge2\), the condition \[ \alpha<1+\frac{2}{n} \] matches the classical Fujita threshold numerically, but the behavior is reversed by the nonlocal feedback: in the local problem this is the blow-up regime, whereas in the nonlocal problem the paper proves global bounded solutions for all nonnegative initial data under the corresponding structural condition [1510.07832]. The paper emphasizes that “by switching on the nonlocal effect” the solution’s behavior changes “from finite time blow-up to global existence” [1510.07832]. ## 5. Forcing, damping, and interface effects Spatial or temporal forcing can substantially modify the critical threshold. For the Hardy–Hénon equation with forcing \[ u_t+(-\Delta)^d u=|x|^\alpha |u|^p+\zeta(t)\mathbf{w}(x), \] the Fujita exponent depends on the time exponent \(\sigma\) when \(\zeta(t)=t^\sigma\): \[ p_F(\sigma)=\frac{N-2d\sigma+\alpha}{N-2d\sigma-2d}\qquad \text{for }-1<\sigma<0, \] and \(p_F(\sigma)=\infty\) for \(\sigma\ge0\) or \(\sigma\le-1\) in the forced setting considered there [2204.00259]. This is a precise example in which the critical exponent depends simultaneously on the diffusion order \(d\), the Hardy–Hénon weight \(\alpha\), the space dimension \(N\), and the temporal forcing exponent \(\sigma\) [2204.00259]. Scale-invariant damping can cause a transition from Fujita-type to Kato-type behavior. For the \(\sigma\)-evolution equation with time-dependent damping, \[ u_{tt}+(-\Delta)^\sigma u+\frac{\mu}{1+t}u_t=|u|^p, \] the critical exponent is \[ p_c=1+\max\left\{\frac{2\sigma}{[n-\sigma+\sigma\mu]_+},\frac{2\sigma}{n}\right\}, \] which equals the Fujita-type value \(1+\frac{2\sigma}{n}\) for \(\mu>1\), and a shifted Kato-type value for \(0<\mu<1\) [2008.10374]. This reflects a change in the large-time linear behavior: effective damping yields parabolic decay and a Fujita-type threshold, while non-effective damping leads to a hyperbolic critical exponent [2008.10374]. By contrast, some singular perturbations leave the Fujita exponent unchanged. For the semilinear parabolic equation with interface drift, \[ \partial_t u=\Delta u+2\mathfrak{q}\,\delta_{\mathbb{S}}\nabla u+|u|^{p-1}u, \] the critical exponent remains \[ p_F=1+\frac{2}{N}, \] the classical Fujita exponent for the heat equation [2606.28248]. The paper shows finite-time blow-up for \(11+\frac{2}{N}\), indicating that the Fujita phenomenon is stable under this discontinuous diffusion effect and interface transmission condition [2606.28248]. ## 6. Methods, critical spaces, and related notions Across these settings, the proofs typically combine linear semigroup estimates with nonlinear iteration, comparison, or test-function methods. Gaussian or Gaussian-type heat kernel bounds are central when the kernel is not explicit, as in weighted degenerate operators, Hörmander sums of squares, and interface problems [2212.12491], [2511.04196], [2606.28248]. In stratified or subelliptic settings, the homogeneous dimension enters through heat kernel decay and ball-volume growth [2211.10759], [1908.02989]. On the global-existence side, mild formulations and contraction arguments in critical or weak spaces are standard. In the weighted degenerate problem, the small-data theory is built in \(L^\infty\cap L^{r^*,\infty}(w)\), where \[ r^*=\frac{n+\alpha}{p-1}, \] and solutions satisfy decay estimates in weighted Lorentz spaces [2212.12491]. In the Hardy–Hénon forcing problem, the critical data space is \[ L^{p_c,\infty}(\mathbb{R}^N),\qquad p_c=\frac{N(p-1)}{2d+\alpha}, \] and the forcing is measured in a second critical Lorentz space indexed by \[ \ell=\frac{N p_c}{N+2(\sigma+1)d\,p_c} \] [2204.00259]. In the Riesz-potential problem, the scale-invariant exponent \[ q_{\mathrm{sc}}=\frac{n(p-1)}{\beta+\alpha} \] still governs the small-data existence space, even though the true Fujita exponent is not the scaling exponent [2512.04506]. On the blow-up side, test-function and nonlinear capacity methods recur throughout the literature. They are used in the Hardy–Hénon setting [2204.00259], on the Heisenberg group [2207.03744], in the Riesz-potential problem [2512.04506], and for equations with interfaces [2606.28248]. Many of these arguments derive contradictions by testing against space–time cutoffs adapted to the underlying scaling, then sending the cutoff radius or time horizon to infinity. The notion of Fujita exponent also interacts with related critical exponents. In the fractional diffusion paper, the stationary Liouville exponent \[ p_{sg}=1+\frac{\alpha}{n-\alpha} \] appears alongside the parabolic Fujita exponent \(1+\frac{\alpha}{n}\) [1706.01251]. In the sublinear Fujita problem, a transitional stability exponent \[ p_c(N)=\frac{N}{N+2} \] is identified, satisfying the reciprocity relation \[ p_F(N)\,p_c(N)=1, \] where \(p_F(N)=1+\frac{2}{N}\) is the classical Fujita exponent [2411.07437]. This suggests that Fujita-type criticality is part of a broader family of threshold phenomena governing not only blow-up versus global existence but also stability versus instability. ## 7. Conceptual synthesis The Fujita exponent is best understood as a semigroup-determined critical threshold rather than a purely formal scaling quantity. In the classical heat equation it is \(1+\frac{2}{N}\) [1510.07832]. In fractional diffusion it becomes \(1+\frac{\alpha}{n}\) or \(1+\frac{2s}{N}\), depending on the operator [1706.01251], [2408.01577]. In weighted and degenerate media it depends on an effective dimension such as \(n+\alpha\) [2212.12491]. On stratified Lie groups and the Heisenberg group it is governed by the homogeneous dimension \(Q\) or \(\mathscr{Q}\) [2211.10759], [1908.02989], [2207.03744]. With forcing, it can depend on temporal decay rates or Hardy–Hénon weights [2204.00259]. With genuinely nonlocal nonlinearities, it may cease to coincide with the scaling exponent altogether [2512.04506]. A plausible implication is that the term “Fujita exponent” now denotes not a single number but a structural role: the sharp boundary in power-law nonlinearities where linear dispersal, diffusion, or damping ceases to control positive solutions globally. What remains invariant across the literature is the dichotomy itself. What changes is the effective geometry, diffusion order, kernel tail, or forcing law that enters the critical formula. Source: https://www.emergentmind.com/topics/fujita-exponent