The first repeating fast radio burst ( FRB ) , FRB 121102 , was found to be associated with a spatially coincident , persistent nonthermal radio source , but the origin of the persistent emission remains unknown .
In this paper , we propose that the persistent emission is produced via synchrotron-heating process by multiple bursts of FRB 121102 in a self-absorbed synchrotron nebula .
As a population of bursts of the repeating FRB absorbed by the synchrotron nebula , the energy distribution of electrons in the nebula will change significantly .
As a result , the spectrum of the nebula will show a hump steadily .
For the persistent emission of FRB 121102 , the total energy of bursts injecting into the nebula is required to be about 3.3 \times 10 ^ { 49 } \mathrm { erg } , the burst injection age is over 6.7 \times 10 ^ { 4 } \mathrm { yr } , the nebula size is \sim 0.02 \mathrm { pc } , and the electron number is about 3.2 \times 10 ^ { 55 } .
We predict that as more bursts inject , the brightness of the nebula would be brighter than the current observation , and meanwhile , the peak frequency would become higher .
Due to the synchrotron absorption of the nebula , some low-frequency bursts would be absorbed , which may explain why most bursts were detected above \sim 1 ~ { } \mathrm { GHz } .