Adaptive streaming based on DASH offers personalized video experience and smooth playback by allowing dynamical adjustments of the video bitrate to the variations of network conditions. This is especially important for current and future Internet video streaming applications, including emerging ones such as virtual reality-based, as adaptive streaming plays a key role in providing high quality viewing experience, especially in limited bandwidth delivery environments. To enable this promising avenue in a 5G context, efforts are made to consider it alongside multicast and edge caching, as part of the next generation communication technology. In this paper, we model the adaptive streaming transmission problem in a mobile scenario as a multi-source multicast multi-rate problem (MMMP) whose linear relaxation is concave. We decompose the problem in terms of clients and propose the distributed delivery algorithm (DDA). The computation complexity, convergence and time-varying adaptation of DDA are theoretically analyzed. Additionally, to further reduce the computation complexity of the solution, a heuristic approximation method (H-DDA) based on the physical meaning of the problem is proposed and it is also shown how H-DDA converges to the optimal value by numerical means. Finally, we conduct a series of simulation tests to demonstrate the superiority of the proposed HDDA in comparison with other state-of-art solutions.