Evolution of thick domain walls in de Sitter universe

We consider thick domain walls in a de Sitter universe following paper by Basu and Vilenkin. However, we are interested not only in stationary solutions found therein, but also investigate the general case of domain wall evolution with time. When the wall thickness parameter, $\delta_0$, is smaller than $H^{-1}/\sqrt{2}$, where $H$ is the Hubble parameter in de Sitter space-time, then the stationary solutions exist, and initial field configurations tend with time to the stationary ones. However, there are no stationary solutions for $\delta_0 \geq H^{-1}/\sqrt{2}$. We have calculated numerically the rate of the wall expansion in this case and have found that the width of the wall grows exponentially fast for $\delta_0 \gg H^{-1}$. An explanation for the critical value $\delta_{0c} = H^{-1}/\sqrt{2}$ is also proposed.