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Compared to periodic systems, quasicrystals without translational invariance exhibit unexpected localization properties. The extended-localized transition in quasicrystals has been observed in both quantum and classical wave systems. However, its manifestation in diffusion systems, which serve as novel platforms for exploring phases of matter in condensed matter physics, remains unexplored. Here, we present the implementation of the extended-localized transition in a diffusive quasicrystal based on the coupled ring chain structure. By modulating the thermal conductivities of rings, we obtain the diffusive one-dimensional Aubry-André-Harper (AAH) model, which exhibits an extended-localized transition. Thanks to the ring-shaped chain, we clearly demonstrate the extended-localized transition under the uniform excitation through temperature field simulations. For the localized state, the temperature field clearly demonstrates a multiple localization centers phenomenon, which has no counterpart in wave systems. We also quantitatively investigate the temperature evolution and size effect of this transition. Furthermore, the local excitation has been adopted to demonstrate the temperature field for both the extended and localized states. Besides, we implement the non-Hermitian diffusive AAH model by rotating rings, whose temperature field shows a moving multiple localization centers phenomenon in the localized phase. Finally, we give the experimental suggestions for the diffusive AAH model and propose a potential application named as double-trace distributed generator. Our results can facilitate the design of flexible thermal devices and efficient heat management.