Hand grasp recognition with surface electromyography (sEMG) has been used as a possible natural strategy to control hand prosthetics. However, effectively performing activities of daily living for users relies significantly on the long-term robustness of such recognition, which is still a challenging task due to confused classes and several other variabilities. We hypothesise that this challenge can be addressed by introducing uncertainty-aware models because the rejection of uncertain movements has previously been demonstrated to improve the reliability of sEMG-based hand gesture recognition. With a particular focus on a very challenging benchmark dataset (NinaPro Database 6), we propose a novel end-to-end uncertainty-aware model, an evidential convolutional neural network (ECNN), which can generate multidimensional uncertainties including vacuity and dissonance, for robust long-term hand grasp recognition. To avoid heuristically determining the optimal rejection threshold, we examine the performance of misclassification detection in the validation set. Extensive comparisons of accuracy under the non-rejection and rejection scheme are conducted when classifying 8 hand grasps (including rest) over 8 subjects across proposed models. The proposed ECNN is shown to improve the recognition performance, achieving an accuracy of 52.44% without the rejection option and 83% under the rejection scheme with multidimensional uncertainties, which significantly improves the current state-of-the-art (SoA) by 5.72% and 13.19%, respectively. Furthermore, its overall rejection-capable recognition accuracy remains stable with only a small accuracy degradation after the last data acquisition over 3 days. These results show the potential design of a reliable classifier that yields accurate and robust recognition performance.