Osgood-type barriers for fractional differential equations with regularly varying coefficients

We study finite-time blow-up phenomena for nonlinear fractional differential equations with superlinear source terms and time-dependent coefficients. For 0 < α < 1 and p > 1 , we consider both the Riemann–Liouville and Caputo models D α u ( t ) = λ ( t ) u ( t ) p + f ( t ) , where λ ≥ 0 is measurable and locally integrable on ( 0 , ∞ ) , while f ≥ 0 is locally integrable. We extend the barrier method recently introduced by Webb [J. R. L. Webb, Electron. J. Differential Equations 2024, No.~40, 1--16] and derive a sharp Osgood-type integral criterion for blow-up. In both fractional settings, this criterion provides an explicit upper bound for the maximal interval of existence and, in particular, rules out global nontrivial solutions whenever the condition is satisfied. We then apply the general framework to coefficients of regularly varying type λ ( t ) = t β ℓ ( t ) , where ℓ is slowly varying, and more broadly to the logarithmic-order class M ( β ) . In the subcritical regime, the relevant barrier integral diverges with the same critical exponent p 0 = 1 + β 1 − α as in the pure power case. At the critical exponent, however, the qualitative behavior depends on the borderline integrability of ℓ ( s ) s , or on finer oscillatory effects within M ( β ) . Several examples are included to illustrate the sharpness of the results and the subtle character of the critical threshold.