Critical equality for polynomial reaction coefficients

Determine whether the equality case p=1+\frac{\beta(q+1)}{d} separates finite-time blow-up from small-data global existence for semilinear heat equations with power nonlinearity f(u)=u^p, time coefficient h(t)=(1+t)^q with q>-1, heat-kernel decay governed by exponent \beta, and polynomial volume growth of order d.

Background

The paper derives finite-time blow-up below the threshold p=1+\frac{\beta(q+1)}{d} and small-data global existence above it under corresponding lower and upper heat-kernel assumptions. For the autonomous case q=0, an additional critical-mass argument resolves the equality case in favor of blow-up. For general q>-1, however, the asymptotic criteria presented in the paper leave the equality case unresolved.

References

For general $q>-1$, the preceding asymptotic criteria do not settle the equality case.

— Semigroup Criteria for Blow-up and Global Existence of Semilinear Heat Equations on Metric Measure Spaces  (2610.01588 - Meglioli et al., 1 Oct 2026) in Section 5, subsection “Power nonlinearities: the role of the time coefficient,” Example “Global existence under polynomial heat kernel decay”