Autonomous linear-slope spectral threshold

Determine the behavior of sufficiently small initial data at the autonomous threshold a=\lambda_1 for nonlinearities with positive linear slope a=f'(0), including f(u)=a_0u+bu^p and f(u)=c(e^{bu}-1), when \lambda_1 is the bottom of the spectrum.

Background

For autonomous reactions h\equiv1 with positive linear slope a, the paper establishes small-data global existence when a<\lambda_1 and finite-time blow-up when a>\lambda_1, under the stated heat-kernel assumptions. The strict inequalities leave the threshold a=\lambda_1 unresolved; the same issue is reiterated in the discussion of regular discrete trees.

References

The autonomous borderline $a=\lambda_1$ is not settled by these strict criteria.

— 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 Proposition 6.4, “Time coefficients in the presence of a linear part”