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

We study the lifespan of mild solutions to the fractional semilinear parabolic Cauchy problem with a Hardy–Hénon-type weight $$u_t+(-\mathrm{\Delta }{)}^{s}u=\mathrm{\mid }x{\mathrm{\mid }}^{-\gamma }\mathrm{\mid }u{\mathrm{\mid }}^p,(t,x)\in (0,\mathrm{\infty })\times {\mathbb{R}}^N,u(0,x)=\epsilon u_0(x),$$ where $0<s<1$, $0\le \gamma <\min (2s,N)$, $p>1$ and $u_0\in L^1\cap L^{\mathrm{\infty }}$ with ${\int }_{{\mathbb{R}}^N}{u}_0(x)dx>0$. Setting $$p_F:=1+\frac{2s-\gamma }{N},$$ we prove that the lifespan $T_{\epsilon }$ obeys, for every sufficiently small $\epsilon >0$, $$T_{\epsilon }\approx \{\begin{array}{ll}{\epsilon }^{-{\beta }^{-1}},& 1<p<p_F,\\ \exp \text{\hspace{0.17em}⁣}\left(C{\epsilon }^{-(p-1)}\right),& p=p_F,\\ +\mathrm{\infty },& p>p_F,\end{array}\beta =\frac{(2s-\gamma )-N(p-1)}{2s(p-1)}\mathrm{.}$$ The lower bound rests on fractional heat-kernel estimates and an $L^1$–$L^{\mathrm{\infty }}$ 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 $(-\mathrm{\Delta }{)}^s$ 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.