  We study the non-autonomously forced Burgers equation
  $$
  u_t(x,t) + u(x,t)u_x(x,t) - u_{xx}(x,t) = f(x,t)
  $$ on the space interval $(0,1)$ with two sets of the boundary conditions:
the Dirichlet and periodic ones. For both situations we prove that there exists
the unique $H^1$ bounded trajectory of this equation defined for all $t\in
\mathbb{R}$. Moreover we demonstrate that this trajectory attracts all
trajectories both in pullback and forward sense. We also prove that for the
Dirichlet case this attraction is exponential.
