The separation of training and data transmission as well as the frequent uplink/downlink (UL/DL) switching make time-division duplex (TDD)-based massive multiple-input multiple-output (mMIMO) systems less competent in fast time-varying scenarios due to the resulted severe channel aging. To this end, a multicarrier-division duplex (MDD) mMIMO scheme associated with two types of well-designed frame structures are introduced for combating channel aging when communicating over fast time-varying channels. To compare with TDD, the corresponding frame structures related to 3GPP standards and their variant forms are presented. The MDD-specific general Wiener predictor and decision-directed Wiener predictor are introduced to predict the channel state information, respectively, in the time domain based on UL pilots and in the frequency domain based on the detected UL data, considering the impact of residual self-interference (SI). Moreover, by applying the zero-forcing precoding and maximum ratio combining, the closed-form approximations for the lower bounded rate achieved by TDD and MDD systems over time-varying channels are derived. Our main conclusion from this study is that the MDD, endowed with the capability of full-duplex but less demand on SI cancellation than in-band full-duplex (IBFD), outperforms both the conventional TDD and IBFD in combating channel aging.