Flux-Ratio Theorems for Nonlinear Equations of Generalized Diffusion

Flux-ratio theorems compare the flux of matter through partof the boundary of a medium with spatially inhomogeneous transportproperties incorporating diffusion, migration, and temporarytrapping of the transported substance, with the flux measuredin a complementary experiment and usually through a differentpart of the boundary. Any nonlinearity in the transport equationsleads to a breakdown of the Ussing flux-ratio theorem pertainingto all times. A fluxintegral theorem is proved for the casewhen the nonlinearity is in the kinetics of trapping. New resultson the time-lag constants for the asymptotics of the two complementaryfluxes associated with nonlinear trapping show that one canexpect deviations from the results that would hold if Ussing'sgeneral theorem were true for this case. When the nonlinearityis due both to nonlinear trapping kinetics and to concentrationdependence of the diffusion coefficient, a nonlinear flux-integral-ratiotheorem is shown to hold when the substance diffuses (but doesnot migrate) in a spatially homogeneous medium.