Qualitative theory for a model of laminar flames with arbitrary nonnegative initial data

AbstractWe consider a system of nonlinear parabolic partial differential equations arising as a model of laminar flames in a premixed reactive gas. The existence of traveling wave solutions has been established for positive ignition temperature by Berestycki, Nicolaenko, and Scheurer and by Terman for zero ignition temperature. Stability and instability of these solutions have been established in various situations by Sivashinsky, Clavin, and recently Terman. Our goal is to study the equations with initial data that are bounded, uniformly continuous, and nonnegative but otherwise arbitrary. We establish the existence of unique global strong solutions satisfying appropriate a priori estimates. With a positivity condition imposed on the initial data for the temperature, we show that the concentration decays exponentially. This result, while easy to obtain, plays an important role in results that follow. Of greatest physical interest are the cases where ignition occurs precisely at one end. Our main result is that if the average of the initial temperature values at the ends of the chamber is above ignition temperature, then on any ray coming from the ignition end the temperature is uniformly above ignition temperature, and the concentration decays uniformly to zero. We can continually advance the endpoint of this zone of ignition, thus roughly mimicking the motion of traveling wave solutions. We also point out the appropriateness of the averaging condition on the initial temperature and discuss two examples where this averaging condition is not satisfied: in one, we eventually have flame propagation and in the other example we have eventual flame extinguishment on any ray coming from the cold end of the chamber