Uniform Shadow Limit Reduction for Reaction-Diffusion-ODE Systems

Reaction-diffusion equations coupled with ordinary differential equations (ODEs) are used to model various biological, chemical and ecological processes. In case some diffusion coefficients tend to infinity, the reaction-diffusion-ODE system can be approximated by a reduced system. This system is called \emph{shadow limit} and is used to facilitate model analysis. A convergence result is well-known for time intervals which are finite compared to the large diffusion parameter. This research investigates the relation between a reaction-diffusion-ODE system endowed with zero flux boundary conditions and its shadow limit on long-time scales. Such long-time intervals scale with the diffusion coefficient and tend to infinity as diffusion tends to infinity. Solutions of both systems are compared with respect to the $L^\infty$ norm and errors are estimated in terms of the inverse of the large diffusion parameter. This work shows that an extension of uniform error estimates to large time intervals may fail without additional stability assumptions. Error estimates are derived by using a uniform stability condition for the evolution of the linearized subsystem of ODEs and of the linearized shadow system. The method is based on previous results for short-time intervals which use a cut-off technique applied to the system linearized at the shadow solution. The partial lack of diffusion implies low regularity in space of solutions to both systems. Hence, mild solutions are considered in this work. Moreover, two analytical ways of verifying the stability conditions are discussed in detail: dissipativity of evolution systems and linearized stability of stationary shadow solutions using a spectral analysis. The general framework applied in this thesis allows to study the uniform shadow limit approximation for reaction-diffusion systems and reaction-diffusion-ODE systems, under low regularity of the solutions and of the domain. The explicit error estimates provide information on the long-term dynamics of such models from results obtained for their shadow limit. Additionally, this detailed study shows that the shadow limit reduction exhibits characteristic time scales. Validity of the approximation on these time ranges can be verified under certain stability assumptions on the shadow system