Asymptotic stability of reaction-diffusion systems in chemical reactor and combustion theory

AbstractSome coupled reaction-diffusion systems arising from chemical diffusion processes and combustion theory are analyzed. This analysis includes the existence and uniqueness of positive time-dependent solutions, upper and lower bounds of the solution, asymptotic behavior and invariant sets, and the stability of steady-state solutions, including an estimate of the stability region. Explicit conditions for the asymptotic behavior and the stability of a steady-state solution are given. These conditions establish some interrelationship among the physical parameters of the diffusion medium, the reaction mechanism, the initial function and the type of boundary condition. Under the same set of physical parameters and reaction function, a comparison between the Neumann type and Dirichlet or third type boundary condition exhibits quite different asymptotic behavior of the solution. For the general nonhomogeneous system, multiple steady-state solutions may exist and only local stability results are obtained. However, for certain models it is possible to obtain global stability of a steady-state solution by either increasing the diffusion coefficients or decreasing the size of the diffusion medium. This fact is demonstrated by a one-dimensional tubular reactor model commonly discussed in the literature