Well-Posedness of Flux–Fractional Compartment Models and Their State Sensitivity Systems

We studied flux–fractional compartment models with Caputo memory terms and their associated state sensitivity systems. The state, classical parameter sensitivities, and fractional-order sensitivity are formulated within a unified linear Volterra framework. Using a Sobolev-space formulation and an equivalent Bielecki norm, we establish existence and uniqueness in W1,p(0,T;H) on arbitrary finite time intervals. We further prove continuous differentiability of the parameter-to-solution map and derive a dimensionally consistent fractional-order sensitivity involving a dimensionless logarithmic kernel. The theory is illustrated by a one-compartment model with explicit Mittag–Leffler representations and relative sensitivity analysis.