Asymptotic analysis of transmission problems with parameter-dependent Robin conditions

TL;DR

This paper performs an asymptotic analysis of a transmission problem with parameter-dependent Robin conditions, exploring limits where the parameter approaches zero or infinity, and examines implications for biological cell models and time-dependent permeability.

Contribution

It provides new asymptotic results for Robin transmission problems, linking mathematical analysis with biological interpretations and extending solutions beyond blow-up times.

Findings

Rates of convergence for limits of the parameter

Connection between asymptotics and Mosco convergence

Extension of solutions beyond finite-time blow-up

Abstract

We study a transmission problem of {N}eumann--{R}obin type involving a parameter $\alpha$ and perform an asymptotic analysis with respect to $\alpha$. The limits $\alpha \to 0$ and $\alpha \to +\infty$ correspond respectively to complete decoupling and full unification of the problem, and we obtain rates of convergence for both regimes. Biologically, the model describes two cells connected by a gap junction with permeability $\alpha$: the case $\alpha \to 0$ corresponds to a situation where the gap junction is closed, leaving only tight junctions between the cells so that no substance exchange occurs, while $\alpha \to +\infty$ corresponds to a situation that can be interpreted as the cells forming a single structure. We also clarify the relationship between the asymptotic analysis with respect to the parameter $\alpha$ and the asymptotics of the system in connection with the convergence of convex functionals known as {M}osco convergence. Finally, we consider time-dependent permeability and analyze the case where $\alpha$ blows up in finite time. Under suitable regularity assumptions, we show that the solution can be extended beyond the blow-up time, remaining in the single structure regime.