Borderline spectral balance for exponentially growing reaction coefficients

Determine whether sufficiently small nontrivial initial data for the equation with power nonlinearity f(u)=u^p and reaction coefficient h(t)=e^{qt} generate global solutions at the borderline q=(p-1)\lambda_1, where \lambda_1 is the bottom of the spectrum and the available exponential heat-kernel estimates alone are inconclusive.

Background

For q<(p-1)\lambda_1, the paper proves global existence for sufficiently small data, while for q>(p-1)\lambda_1 it proves finite-time blow-up for every admissible nontrivial datum under the relevant lower heat-kernel assumptions. The borderline balance is not resolved by the exponential estimates alone. The paper notes that an additional polynomial correction in the heat-kernel decay can yield small-data global existence in some borderline cases, but does not settle the general borderline problem.

References

At the borderline $q=(p-1)\lambda_1$, these exponential estimates alone do not decide whether sufficiently small nontrivial data generate global solutions.

— 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 Remark 5.10, “Power nonlinearities: the role of the time coefficient”