With the increasing frequency of extreme climate events, enhancing precipitation forecasting capability has become an urgent need in meteorological operations. Most existing data-driven methods model precipitation motion and intensity in a coupled manner. Although effective at capturing large-scale precipitation systems, these methods struggle to accurately predict the rapid evolution of small- to medium-scale heavy precipitation, thereby limiting their forecast skill for intense rainfall events. In this study, a dual-branch fusion network named LAFUNet is proposed, which decouples motion and intensity via Lagrangian transformation. One branch directly analyzes original radar image sequences to capture the spatial structure and motion characteristics of large-scale precipitation systems. The other branch transforms the precipitation field into the Lagrangian coordinates, focusing on modeling intensity evolution to better represent the nonlinear intensity changes associated with small- to medium-scale heavy precipitation. Additionally, a dual-branch interaction module is designed to adaptively fuse features from both branches. Experiments are conducted on the public CIKM and SEVIR radar datasets. The results demonstrate that the proposed model achieves outstanding performance in heavy precipitation nowcasting. Particularly on the SEVIR dataset, for extreme precipitation events with an intensity threshold exceeding 219, the model attains a CSI score of 0.1368 for 1 hour forecasts, significantly outperforming comparative models such as VMRNN.