In this paper, we present a deep multi-task learning framework able to couple semantic segmentation and change detection using fully convolutional long short-term memory (LSTM) networks. In particular, we present a UNet-like architecture (LUNet) which models the temporal relationship of spatial feature representations using integrated fully convolutional LSTM blocks on top of every encoding level. In this way, the network is able to capture the temporal relationship of spatial feature vectors in all encoding levels without the need to downsample or flatten them, forming an end-to-end trainable framework. Moreover, we further enrich the L-UNet architecture with an additional decoding branch that performs semantic segmentation on the available semantic categories that are presented in the different input dates, forming a multi-task framework. Different loss quantities are also defined and combined together in a circular way to boost the overall performance. The developed methodology has been evaluated on three different datasets, i.e, the challenging bi-temporal high-resolution ONERA Satellite Change Detection (OSCD) Sentinel-2 dataset, the very high-resolution multitemporal dataset of the East Prefecture of Attica, Greece, and lastly, the multitemporal very high-resolution SpaceNet7 dataset. Promising quantitative and qualitative results demonstrated that the synergy among the tasks can boost up the achieved performances. In particular, the proposed multi-task framework contributed to a significant decrease of false positive detections, with F1 rate outperforming other state of the art methods by at least 2.1% and 4.9% in the Attica VHR and SpaceNet7 dataset case respectively. Our models and code can be found at: https://github.com/mpapadomanolaki/multi-task-L-UNet