Qualitative Theory of the Cauchy Problem for a One-Step Reaction Model on Bounded Domains

• Published in: SIAM journal on mathematical analysis Vol. 22; no. 2; pp. 379 - 391
• Main Author: Avrin, Joel D.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.03.1991
• Subjects: Boundary conditions; Cauchy problems; Exact sciences and technology; Heat; Mathematical methods in physics; Physics; Laminar flame; Asymptotic behavior; Boundary condition; Reaction diffusion equation; Burning; Cauchy problem; Quenching
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/0522024

A system of reaction-diffusion equations on a bounded domain arising as a model of laminar flames in a premixed reactive gas is considered. The equations couple the temperature $T$ and the concentration $Y$ subject to Arrhenius kinetics. After establishing the existence of unique global strong solutions for arbitrary nonnegative initial data in $L^p$, the bulk of the paper is devoted to an examination of the qualitative behavior of solutions subject to various boundary conditions; showing in particular that both $T$ and $Y$ remain bounded in most of the cases discussed. In the no-flux case in which both $T$ and $Y$ satisfy zero Neumann boundary conditions, it is shown that if the average of the initial temperature over the domain is larger than ignition temperature, then eventually $T$ is uniformly above ignition temperature and $Y$ eventually decays exponentially to zero. This situation is called complete asymptotic burning, and an example is given to show that it does not always occur if the averaging condition is not met; if $Q$ is the chemical heat release it is in fact shown in the case of equal diffusion coefficients that there exist parameters $Q^ * $ and $Q_ * $ with $Q_ * \leqq Q^ * $ such that complete asymptotic burning or eventual flame quenching occurs if, respectively, $Q > Q^ * $ or $Q < Q^ * $. In this case convergence to constant steady states is also established if $Q > Q^ * $ or $Q < Q_ * $. For $T$ satisfying fixed Dirichlet boundary conditions an example of complete asymptotic burning, and an example of flame quenching in at least portions of the domain are constructed. When $Y$ satisfies fixed Dirichlet boundary condtions, cases are constructed where $Y$ is bounded away from zero in portions of the domain if a certain parameter appearing in the Arrenius rate law is small enough.