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Multiwavelength light curves in long-term campaigns have shown that, forseveral blazars, the radio emission occurs with a significant delay w.r.t. to$\gamma$-ray band, with timescales ranging from weeks to years. Suchobservational evidence has been a matter of debate for years and usually isinterpreted as a signature of the $\gamma$-ray emission originating upstream inthe jet, with the emitting region becoming radio transparent at larger scales.In this paper, we show, by means of self-consistent numerical modelling, thatthe adiabatic expansion of relativistic blob can explain these delays. We usethe JetSeT framework to reproduce the numerical modelling of theradiative/accelerative processes, reproducing the temporal evolution, from theinitial flaring activity, and the subsequent expansion. We follow the spectralevolution and the light curves, investigating the relations among the observedparameters, rise, time, and decay time, identifying the link with the physicalparameters. We find that, when adiabatic expansion is active, lags due to theshift of the synchrotron frequency occur, with an offset equal to the distancein time between the flaring onset and the beginning of the expansion, whilstthe rising and decaying timescales depend on the velocity of the expansion andon time required to the source to exhibit a synchrotron self-absorptionfrequency. We derive an inter-band response function, and we investigate theeffects of the competitions between radiative and adiabatic cooling timescaleson the response. We apply the response function to long-term radio and$\gamma-$ray light curves of Mrk 421, Mrk 501 and 3C 273, finding asatisfactory agreement on the log-term behaviour, and we use a Monte CarloMarkov Chain approach, to estimate some physical relevant parameters. Wediscuss the applications to polarization measurements, and to jets collimationprofile kinematics.

Radio$-\gamma-$ray response in blazars as a signature of adiabatic blob expansion