Limit Set of Trajectories of the Coupled Viscous Burgers' Equations

In this letter, a coupled system of viscous Burgers' equations with zero Dirichlet boundary conditions and appropriate initial data is considered. For the well-known single viscous Burgers' equation with zero Dirichlet boundary conditions, the zero equilibrium is the unique global exponential point attractor. A similar property is shown for the coupled Burgers' equations, i.e., trajectories starting with initial data which is not too large approach the zero equilibrium as time goes to infinity. This ``approaching" or convergence is not necessarily exponentially fast, unlike the single viscous Burgers' equation.