Metastability for a generalized Burgers equation with applications to propagating flame fronts

In the small diffusion limit epsilon--> 0, metastable dynamics is studied for the generalized Burgers problem u(t) + f'(u )u(x) - f'(u) = epsilon u(xx), 0 < x < 1, t > 0 u(0,t) = u(1,t) = 0, u(x,0) = u(0)(x). Here u = u(x,t) and f(u) is smooth, convex, and satisfies f(0) = f'(0) = 0. The choice f(u)= u(2)/2 has been shown previously to arise in connection with the physical problem of upward flame-front propagation in a vertical channel in a particular parameter regime. In this context, the shape y = y(x, t) of the flame-front interface satisfies u = -y(x). For this problem, it is shown that the principal eigenvalue associated with the linearization around an equilibrium solution corresponding to a parabolic-shaped flame-front interface is exponentially small. This exponentially small eigenvalue then leads to a metastable behaviour for the time-dependent problem. This behaviour is studied quantitatively by deriving an asymptotic ordinary differential equation characterizing the slow motion of the tip location of a parabolic-shaped interface. Similar metastability results are obtained for more general f(u). These asymptotic results are shown to compare very favourably with full numerical computations.