Exponential stability of the flow for a generalised Burgers equation on a circle

The paper deals with the problem of stability for the flow of the 1D Burgers equation on a circle. Using some ideas from the theory of positivity preserving semigroups, we establish the strong contraction in the $L^1$ norm. As a consequence, it is proved that the equation with a bounded external force possesses a unique bounded solution on $\mathbb{R}$, which is exponentially stable in $H^1$ as $t\to +\mathrm{\infty }$. In the case of a random external force, we show that the difference between two trajectories goes to zero with probability $1$.