Decay and boundedness results for a model of laminar flames with complex chemistry

Abstract:

We consider the reaction-diffusion equations modeling two-step reactions with Arrhenius kinetics on bounded spatial domains or over all of ${\mathbb {R}^n}$. After noting the existence, uniqueness, and nonnegativity of global strong solutions with virtually arbitrary nonnegative initial data, we give conditions on the initial temperature that guarantee decay of the concentrations to zero and a supremum norm bound on the temperature. In our first such result we assume that the initial temperature ${T_0}$ is uniformly bounded above the two ignition temperatures. Specializing to the case of bounded spatial domains, we replace this condition by the more general requirement that the average of ${T_0}$ over the domain is above both ignition temperatures. Finally, we note a boundedness result with equal diffusion coefficients that holds for arbitrary choices of the other parameters. Combining this assumption with the hypotheses, noted above, about the initial temperature, we obtain steady-state convergence results for the temperature as well as the concentrations.