Burgers' equation with time delay is considered. Using the Cole-Hopf transformation, the exact solution of this nonlinear partial differential equation (PDE) is determined in the context of a (seemingly) well-posed initial-boundary value problem (IBVP) involving homogeneous Dirichlet data. The solution obtained, however, is shown to exhibit a delay-induced instability, suffering blow-up in finite-time.
Jordan, P. M. (2008). A note on Burgers’ equation with time delay: Instability via finite-time blow-up. Physics Letters, Section A: General, Atomic and Solid State Physics, 372(42), 6363–6367. https://doi.org/10.1016/j.physleta.2008.08.049