Abstract

We consider the generalized Burgers equation with and without a time delay when the boundary conditions are periodic with period 2π. For the generalized Burgers equation without a time delay, that is, ut=vuxx−uux+u+h(x), 0<x<2π, t>0, u(0,t)=u(2π,t), u(x,0)=u0(x), a Lyapunov function method is used to show boundedness and uniqueness of a steady state solution and global stability of the equation. As for the generalized time-delayed Burgers equation, that is, ut(x,t)=vuxx(x,t)−u(x,t−τ)ux(x,t)+u(x,t), 0<x<2π, t>0, u(0,t)=u(2π,t), t>0, u(x,s)=u0(x,s), 0<x<2π, −τ≤s≤0, we show that the equation is exponentially stable under small delays. Using a pseudospectral method, we present some numerical results illustrating and reinforcing the analytical results.