The self-interaction of the Alfvén wave associated with the parametric interaction between the Alfvén wave and longitudinal plasma motion induced by this Alfvén wave is investigated in a dissipative plasma with a thermal misbalance. The thermal misbalance caused by plasma heating and radiation cooling leads to changes in the steepening rate and the resulting amplitude of Alfvén waves. The characteristic times of steepening and changes of amplitude are obtained analytically and subsequently calculated for the initial stage of self-interaction of sinusoidal Alfvén waves applying the upper chromosphere and coronal hole conditions. Comparison with characteristic times of viscous and Ohmic dissipation is conducted as well. It is revealed that for the chromosphere conditions there are temperature ranges where Alfvén waves can steepen faster and, therefore, the nonlinear Alfvén dissipation observed by Grant et al. (Nat. Phys.14(5), 480, 2018) could be stronger. Moreover, depending on the temperature range, the steepening of the wave can occur both in front and behind. In addition, temperature ranges were found in which the Alfvén wave can be strongly absorbed due to the thermal misbalance. It should be mentioned that regions of fast steepening and fast damping due to the thermal misbalance intersect. Therefore, in these regions, the damping of Alfvén waves can be most significant. We also show that for coronal hole conditions the thermal misbalance only affects the Alfvén wave steepening.