  Fast blue optical transients (FBOTs) represent a new class of highly
energetic sources observed from radio to X-rays. High luminosity, light curves
and spectra of the sources can be understood if they are associated with
supernova-like or tidal disruption events. Radio observations of the transient
sources revealed a mildly relativistic expansion of some of the remnants. The
high power and mildly relativistic shock velocities are providing favorable
conditions for very high energy particle acceleration. In this paper we present
a model of particle acceleration in mildly relativistic magnetohydrodynamic
(MHD) outflow of the transient source. To construct the non-thermal radiation
and cosmic ray spectra in a broad range of energies we combined the microscopic
particle-in-cell (PIC) simulations of electron and proton injection at mildly
relativistic shock with Monte Carlo technique for high energy particle
transport and acceleration. The kinetic PIC simulations provided the energy
partition parameter $\epsilon_{e}$ used to fit the observed non-thermal radio
emission using the magnetic field amplification mechanisms modelled with Monte
Carlo simulations. The model allowed to describe the radio-spectrum of
CSS161010 and it's X-ray luminosity. The high X-ray luminosity of AT2018 and
AT2020mrf detected during the first weeks can be connected to the jet
interaction with the stellar companion in a binary system. The model predicts
that FBOTs can accelerate cosmic rays to energies above 10 PeV with a possible
upper limit of maximum energy of 100 PeV. With the expected event rate of FBOTs
they can contribute to the very high energy cosmic rays population in galaxies.
