Quenching for a one-dimensional fully nonlinear parabolic equation in detonation theory

We study and describe the quenching phenomenon for the fully nonlinear para bolic equation u(t) + 1/2 (u(x))(2) = f(cu u(xx)) + ln u, x is an element of (0, l), t >0, which for f (s) = ln [(e(s) - 1)/s] represents the evolution of the perturb ations of the Zel'dovich-von Neuman-Doering square wave occurring during a detonation in a duct. In the general case, the function f : R --> R is smoo th and satis es the parabolicity condition f '(s) > 0 in R c and l are posi tive constants, and we impose Neumann boundary conditions u(x) (0, t) = u(x ) (l, t) = 0 for t> 0 and take initial data u (x, 0) > 0 with inverse bell- shaped form. The phenomenon of quenching is characterized by the existence of a finite t ime T at which the solution u ceases to exist as a classical solution becau se min(x) u(x, t) --> 0 as t T; then the equation degenerates and forms a s ingularity at the level u = 0 due to the presence of the logarithmic zero-o rder term. We first exhibit conditions on f and u (x, 0) which imply the presence of t his type of singularity. Next we derive estimates on u, u(xx) in order to s tudy the behavior of the profile in the neighborhood of the time T. We then nd the asymptotic scaling factors, which are universal, and the asymptotic pro le which is given in the rescaled coordinates by a parabola with a fre e constant to adjust. For this purpose we use the theory of stability of om ega -limit sets of infinite-dimensional dynamical systems under asymptotica lly small perturbations. In this problem the perturbation is singular but e xponentially vanishing as t --> T. Finally we prove that the present model does not admit any extension beyond the singularity i. e., for t > T.