  The dynamics of open quantum systems connected with several reservoirs
attract great attention due to its importance in quantum optics, biology,
quantum thermodynamics, transport phenomena, etc. In many problems, the Born
approximation is applicable which implies that the influence of the open
quantum system on the reservoirs can be neglected. However, in the case of a
long-time dynamics or mesoscopic reservoir, the reverse influence can be
crucial. In this paper, we investigate the transient dynamics of several
bosonic reservoirs connected through an open quantum system. We use an
adiabatic approach to study the temporal dynamics of temperatures of the
reservoirs during relaxation to thermodynamic equilibrium. We show that there
are various types of temperature dynamics that strongly depend on the values of
dissipative rates and initial temperatures. We demonstrate that temperatures of
the reservoirs can exhibit non-monotonic behavior. Moreover, there are moments
of time during which the reservoir with initially intermediate temperature
becomes the hottest or coldest reservoir. The obtained results pave the way for
managing energy flows in mesoscale and nanoscale systems.
