A Stability Analysis for a Fully Nonlinear Parabolic Problem in Detonation Theory

This work is concerned with the the stability analysis of the constant stationary solution of the following fully nonlinear parabolic equation: ut+ 1 2u 2 x = f(cuuxx)+lnu, x ∈ (0, l) with ux(0, t) = ux(l, t) = 0, where f is a smooth function satisfying f(0) = 0, f ′ > 0 and f(IR) = IR. In the case where f(s) = ln [ exp(s)−1 s ] , this equation represents the evolution of the perturbations of the Zeldovich-von Neuman-Doering square wave occuring during a detonation in a duct. We first study the stationary solutions and reveal a bifurcation phenomenon. Then, by formulating the problem as an abstract equation defined on a suitable Banach space, we are able to use the extension to fully nonlinear problems of the classical geometric theory for semilinear parabolic equations. In this way, we prove that the equilibrium point u0 = 1 is unstable. Moreover, a more careful description of a special class of initial conditions for which u0 is stable.