On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

Abstract

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.