Stability and attractors for the quasi-steady equation of cellular flames

Abstract

We continue the study a simple integro-differential equation: the Quasi-Steady equation (QS) of flame front dynamics. This equation is dynamically similar to the Kuramoto-Sivashinsky (KS) equation. In \cite{FGS03}, where it was introduced, its well-posedness and proximity for finite time intervals to the KS equation in Sobolev spaces of periodic functions were established. Here we demonstrate that QS possesses a universal absorbing set, and a compact attractor. Furthermore we show that the attractor is of a finite Hausdorff dimension, and give an estimate on it. We discuss relationship with the Kuramoto-Sivashinsky and Burgers-Sivashinsky equations.