Abstract. Alfvén waves are the most promising candidate to carry energy into the solar corona and, therefore, heat it. However, for this to happen, two contradictory conditions should be satisfied. On one hand, Alfvén waves should propagate over long distances from the lower parts to the upper parts of the solar atmosphere. On the other hand, their magnetic energy should be thermalised to heat the corona. In this study, the various aspects of the shear/torsional Alfvén waves propagation have been considered in the weakly nonlinear limit to address these contradictions. It has been shown that the presence of non-adiabatic processes in plasma can significantly affect the nonlinear dynamics of Alfvén waves. For example, in isentropically unstable plasma, Alfvén waves amplitude grows with time while its profile steepens faster than in the case of ideal plasma. In the opposite case of isentropically stable plasma, Alfvén waves amplitude decreases with time and the wave profile steepens slower.

Moreover, for ideal plasma, it has been shown analytically and numerically that torsional Alfvén waves can induce radially uniform perturbations of density independently of the radial profile of the driver. This finding can be an observational signature of torsional Alfvén waves in the solar corona. However, it requires further justification.