Sharp Lifespan Estimates and Fujita Phenomena for Fractional Hardy-Hénon Type Parabolic Equations

Technical Summary: Sharp Lifespan Estimates and Fujita Phenomena for Fractional Hardy–Hénon Type Parabolic Equations

Problem Statement
The paper investigates the Cauchy problem for the fractional semilinear parabolic equation with a Hardy–Hénon-type weight:
$$\begin{cases} u_t + (-\Delta)^s u = |x|^{-\gamma} |u|^p, & (t, x) \in (0, \infty) \times \mathbb{R}^N, \\ u(0, x) = \varepsilon u_0(x), & x \in \mathbb{R}^N, \end{cases}$$
where $0 < s < 1$, $0 \leq \gamma < \min(2s, N)$, $p > 1$, and the initial data $u_0 \in L^1(\mathbb{R}^N) \cap L^\infty(\mathbb{R}^N)$ satisfies $\int_{\mathbb{R}^N} u_0 \, dx > 0$. The parameter $\varepsilon > 0$ is small. The primary objective is to determine the lifespan $T_\varepsilon$ (the maximal existence time) of mild solutions and to characterize the blow-up behavior, specifically identifying the critical Fujita exponent and establishing sharp lifespan estimates in subcritical, critical, and supercritical regimes.

Methodology
The authors employ a combination of semigroup theory, heat-kernel estimates, and refined test-function methods adapted for nonlocal operators.

1. Lower Bounds (Bootstrap Argument):

   • The lower bound on the lifespan is derived using a bootstrap argument based on an auxiliary functional $M(t)$, which combines the $L^1$ and weighted $L^\infty$ norms.
   • A key tool is a weighted interpolation inequality (Lemma 2.2) that controls the nonlinear term $\||x|^{-\gamma}|u|^p\|_{L^1}$ by interpolating between $L^1$ and $L^\infty$ norms.
   • The proof utilizes sharp $L^p$–$L^q$ smoothing estimates of the fractional heat semigroup $\{e^{-t(-\Delta)^s}\}$ and the specific decay properties of the fractional heat kernel $G_s(t, x)$.

2. Upper Bounds (Test-Function Method):

   • The upper bound is established by testing the equation against the backward fractional heat kernel $G_s(b-t, \cdot)$.
   • Crucially, this approach avoids the use of compactly supported cutoff functions, which are incompatible with the nonlocal nature of $(-\Delta)^s$ (as the operator of a compactly supported function is never compactly supported).
   • By utilizing a globally defined, polynomially decaying test function, the linear terms cancel identically due to the self-adjointness of $(-\Delta)^s$. This reduces the problem to a scalar differential inequality for the functional $g_b(t) = \int u(x,t) G_s(b-t, x) \, dx$.
   • For the critical case ($p = p_F$), the authors employ a scale-superposition technique (integrating over a dyadic band of scales) to recover the logarithmic lifespan behavior.

3. Blow-Up Rate and Profiles:

   • The blow-up rate is analyzed using a nonlocal maximum-principle inequality and scaling arguments.
   • The asymptotic behavior is studied via self-similar variables, leading to a stationary problem involving the fractional Laplacian and an Ornstein–Uhlenbeck drift term.

Key Contributions and Results

1. Sharp Lifespan Estimates (Theorem 1.1):
   The paper establishes the precise lifespan $T_\varepsilon$ for small $\varepsilon$:

   • Subcritical ($1 < p < p_F$): $T_\varepsilon \approx \varepsilon^{-1/\beta}$, where $\beta = \frac{(2s-\gamma) - N(p-1)}{2s(p-1)}$.
   • Critical ($p = p_F$): $T_\varepsilon \approx \exp(C \varepsilon^{-(p-1)})$.
   • Supercritical ($p > p_F$): $T_\varepsilon = +\infty$ (global existence).
     Here, the critical Fujita exponent is identified as $p_F = 1 + \frac{2s-\gamma}{N}$. The exponent $\beta$ is shown to be sharp.

2. Large-Data Lifespan (Theorem 6.1):
   For non-decaying initial data ($u_0 \geq c_0 > 0$), the paper derives a different lifespan law: $T_\varepsilon \approx \varepsilon^{-\frac{2s(p-1)}{2s-\gamma}}$. This contrasts with the small-data regime, highlighting the dependence of the lifespan on the spatial decay of the initial data.

3. Blow-Up Rates (Theorem 7.1):

   • Universal Lower Bound: Unconditionally, any blowing-up nonnegative solution satisfies $N(t) \geq C(T-t)^{-\vartheta}$, where $\vartheta = \frac{2s-\gamma}{2s(p-1)}$. This is the self-similar rate.
   • Interior Lower Bound: If blow-up occurs away from the origin ($|x| \geq \delta$), the rate improves to the ODE rate $N(t) \geq C(T-t)^{-1/(p-1)}$.
   • Conditional Upper Bound: A Type-I upper bound $N(t) \leq C(T-t)^{-\vartheta}$ holds provided the solution satisfies a specific a priori self-similar bound. This condition is linked to the exclusion of Type-II blow-up.

4. Asymptotic Profiles (Theorem 8.1):
   Under the assumption of a Type-I bound, the solution converges (along a subsequence) to a bounded distributional solution of the stationary equation:
   $$(-\Delta)^s V + \frac{1}{2s} y \cdot \nabla V + \vartheta V = |y|^{-\gamma} V^p.$$
   The uniqueness of this profile is conditional on a Liouville-type classification for this stationary problem, which the authors note is currently only partially understood.

Significance and Claims
The authors claim that these results are new even for the case $\gamma = 0$ (the fractional Fujita equation), providing the first sharp two-sided lifespan estimates for the fractional Lee–Ni exponent in this context.

• Correction of Scaling Intuition: The paper explicitly refutes the "natural guess" that the critical exponent or lifespan rate could be obtained by simply replacing $2s$ with $2s-\gamma$ in classical formulas. The authors demonstrate that the Hardy weight enters the integrability of the nonlinearity additively (via the interpolation inequality), leading to the specific additive correction in the exponent $\beta$, rather than a multiplicative modification of the diffusion parameter.
• Methodological Innovation: The use of the backward fractional heat kernel as a test function is highlighted as a significant advancement. It circumvents the limitations of the classical test-function method (which relies on compact support) when dealing with nonlocal operators, allowing for a unified treatment of the upper bounds without artificial cutoffs.
• Regime Distinction: The work clarifies the dichotomy between blow-up at the origin (governed by the singular weight and the slower self-similar rate $\vartheta$) and blow-up in the interior (governed by the local ODE dynamics and the faster rate $1/(p-1)$).

The paper concludes by noting that while the lower bounds and lifespan estimates are unconditional, the matching upper bounds and the uniqueness of the blow-up profile remain conditional on a fractional Liouville classification for the associated stationary problem, which is identified as an open problem.