Damped Burgers equation ($u_t+u{u}_x+\gamma u=\epsilon u_{xx}$)

The Burgers' equation with constant linear damping has the displayed form. For zero viscosity and $\gamma >0$, characteristics are $u=u_0(a)e^{-\gamma t}$ and $x=a+u_0(a)(1-e^{-\gamma t})/\gamma$. A negative initial slope can make this coordinate map cease to be invertible, producing shock formation by characteristic intersection. The transformation $u=e^{-\gamma t}v$, $\tau =(1-e^{-\gamma t})/\gamma$ converts the inviscid equation to the ordinary inviscid Burgers' equation. Across a thin viscous layer the damping contributes only at lower order to the leading Rankine-Hugoniot condition.