This paper addresses the classification of locally conformally flat gradient Yamabe solitons. In the first part it is shown that locally conformally flat gradient Yamabe solitons with positive sectional curvature are rotationally symmetric. In the second part the classification of all radially symmetric gradient Yamabe solitons is given and their correspondence to smooth self-similar solutions of the fast diffusion equation on Rn is shown. In the last section it is shown that any eternal solution to the Yamabe flow with positive Ricci curvature and with the scalar curvature attaining an interior space-time maximum must be a steady Yamabe soliton.
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