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Natural disasters caused by heavy rainfall often cause huge loss of life and property. Hence, the task of precipitation nowcasting is of great importance. To solve this problem, several deep learning methods have been proposed to forecast future radar echo images, and then the predicted maps are converted to the distribution of rainfall. The prevailing spatiotemporal sequence prediction methods apply a ConvRNN structure, which combines the convolution and recurrent neural network. Although ConvRNN methods achieve remarkable success, they do not capture both local and global spatial features simultaneously, which degrades the nowcasting in regions of heavy rainfall. To address this issue, we propose a Region Attention Block (RAB) and embed it into ConvRNN to enhance forecasting in the areas with heavy rainfall. Besides, the ConvRNN models find it hard to memorize longer historical representations with limited parameters. To this end, we propose a Recall Attention Mechanism (RAM) to improve the prediction. By preserving longer temporal information, RAM contributes to the forecasting, especially in the moderate rainfall intensity. The experiments show that the proposed model, Region Attention Predictive Network (RAP-Net), significantly outperforms state-of-the-art methods. [ABSTRACT FROM AUTHOR]