The authors study a one-dimensional viscous Burgers equation \[ u_t(x, t) + u(x, t)u_x(x, t)- u_{xx}(x, t) = f(x, t),\; x\in(0,1) \] with a non-homogeneous and nonautonomous forcing term \(f\in L^\infty(\mathbb{R},L^2(0,1))\).
Under Dirichlet boundary conditions they prove the existence of a pullback attractor consisting of a unique eternal solution, i.e., they show that there exists precisely one globally defined bounded solution \(u(t)\) and that for bounded sets \(B\in L^2(0,1)\) of initial conditions one has \[ \lim\limits_{t_0\to -\infty}\mathrm{dist}_{H^1_0}(S(t, t_0)B, \{u(t)\}) = 0, \] where \(S(t,t_0)\) is the process defined by the partial differential equation.
An analogous result is shown to hold under periodic boundary conditions. These theorems extend known results with time-periodic forcing.
Furthermore, in the Dirichlet case, the authors are able to prove exponential attraction of the eternal solution both in forward time and in the pullback sense.