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The size-dependent exciton dynamics of one-dimensional aggregates of substituted perylene bisimides are studied by ultrafast transient absorption spectroscopy and kinetic Monte-Carlo simulations as a function of the excitation density and the temperature in the range of 25-90 degrees C. For low temperatures, the aggregates can be treated as infinite chains and the dynamics is dominated by diffusion-driven exciton-exciton annihilation. With increasing temperature the aggregates dissociate into small fragments consisting of very few monomers. This scenario is also supported by the time-dependent anisotropy deduced from polarization-dependent experiments.