Sharp Lifespan Estimates and Fujita Phenomena for Fractional Hardy-H\'enon Type Parabolic Equations

We study the lifespan of mild solutions to the fractional semilinear parabolic Cauchy problem with a Hardy--Hénon-type weight \[
u_t + (-\Delta)^s u = |x|^{-\gamma}\,|u|^p,
\qquad (t,x)\in(0,\infty)\times\mathbb{R}^N,
\qquad u(0,x)=\varepsilon\,u_0(x), \] where $0<s<1$, $0\le\gamma<\min(2s,N)$, $p>1$ and $u_0\in L^1\cap L^\infty$ with $\int_{\mathbb{R}^N}u_0(x)\,dx>0$. Setting \[
p_F \;:=\; 1+\frac{2s-\gamma}{N}, \] we prove that the lifespan $T_\varepsilon$ obeys, for every sufficiently small $\varepsilon>0$, \[
T_\varepsilon \;\approx\;
\begin{cases}
\varepsilon^{-\,\beta^{-1}},& 1<p<p_F,\\[1mm]
\exp\!\big(C\,\varepsilon^{-(p-1)}\big),& p=p_F,\\[1mm]
+\infty,& p>p_F,
\end{cases}
\qquad
\beta \;=\;\frac{(2s-\gamma)-N(p-1)}{2s(p-1)}. \] The lower bound rests on fractional heat-kernel estimates and an $L^1$--$L^\infty$ Hardy-type interpolation inequality; the upper bound is obtained by testing the equation against the backward fractional heat kernel, a globally defined positive weight for which $(-\Delta)^s$ is controlled everywhere and the linear terms cancel identically by self-adjointness. This circumvents the compactly supported cutoffs of the classical test-function method, which are incompatible with a nonlocal operator. The exponent $\beta$ is sharp; for $\gamma=0$ it reduces to the fractional Lee--Ni exponent $\frac{1}{p-1}-\frac{N}{2s}$. To the best of our knowledge, these results are new even for $\gamma=0.$ We also establish a large-data lifespan law, sharp lower bounds on the blow-up rate together with a conditional Type-I upper bound, a conditional self-similar profile result.