Nonlinear coupled system in thin domains with corrugated boundaries

In this paper, we study the asymptotic behaviour of solutions of a coupled system of partial differential equations in a thin domain with oscillating boundary and varying order of thickness. In such a thin domain, our model describes the solute concentration of two different biochemical species (metabolites). The coupling between the concentrations of the metabolites is realized through reaction terms even nonlinear, appearing on the oscillating upper wall. Moreover nonlinear reaction terms appear also in the thin domain. The reaction catalyzed by the upper wall is simulated by a Robin-type boundary condition depending on a small parameter $\varepsilon$. Hence, taking into account that $\alpha>1$ and $\beta>0$, we analyze the coupled system by comparing the magnitude of the reaction coefficient $\varepsilon^\beta$ on the upper boundary with the compression order of our thin domain, which can be $\varepsilon$ or $\varepsilon^\alpha$, depending on the sub-regions with different order of thickness. Comparing the exponents $1$, $\alpha$ and $\beta$, we obtain different cases for the limit problem which could appear coupled or uncoupled and allow us to identify the effects of the geometry and the physical process on the problem. Moreover it arises a critical value, i.e. $\beta=\alpha-2$, leading the reaction effects entering in the diffusion matrix.