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

Abstract

We study the lifespan of mild solutions to the fractional semilinear parabolic Cauchy problem with a Hardy--Hénon-type weight \[ u_t + (-Δ)^s u = |x|^{-γ}\,|u|^p, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N, \qquad u(0,x)=\varepsilon\,u_0(x), \] 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 \[ p_F \;:=\; 1+\frac{2s-γ}{N}, \] we prove that the lifespan $T_{\epsilon }$ obeys, for every sufficiently small $\epsilon >0$, \[ T_\varepsilon \;\approx\; \begin{cases} \varepsilon^{-\,β^{-1}},& 1<p<p_F,\\[1mm] \exp\!\big(C\,\varepsilon^{-(p-1)}\big),& p=p_F,\\[1mm] +\infty,& p>p_F, \end{cases} \qquad β\;=\;\frac{(2s-γ)-N(p-1)}{2s(p-1)}. \] 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.