The slow motion of shock layers for advection-diffusion-reaction equations

This paper surveys the work of Laforgue, Knaub, O'Malley and Williams on the long term evolution of a shock layer for singularly perturbed PDEs of the form ut=ε2uxx+εg(u)ux+h(u) as the small, positive parameter ε→0 on a finite x domain with constant boundary values, uR and uL, satisfying h(uR)=h(uL)=0. A traveling wave ansatz based on a limiting shock profile is used. Results when the shock profile has either exponentially or algebraically decaying tails are summarized (cf. [Stud. Appl. Math. 112 (2004) 1-15], [Stud. Appl. Math. 102 (1999) 137-172] and [M.J. Ward, in: Proc. Sympos. Appl. Math., vol. 56, American Mathematical Society, Providence, RI, 1999, pp. 151-184]) and two examples where the profile exhibits algebraically decaying tails are discussed in detail. It is further shown in the case of algebraic asymptotics, analogous to well-known results for exponential asymptotics, that a dynamic metastability occurs, causing the shock profile to slowly drift to a steady state after its initial formation. The algebraic case is also supersensitive to boundary value perturbations of algebraic size, again analogous to the results for the exponential case.