<p>We study a transmission problem of Neumann–Robin type involving a parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and perform an asymptotic analysis with respect to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. The limits <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha \rightarrow +\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> correspond respectively to complete decoupling and full unification of the problem, and we obtain rates of convergence for both regimes. Biologically, the model describes two cells connected by a gap junction with permeability <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>: the case <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> corresponds to a situation where the gap junction is closed, leaving only tight junctions between the cells so that no substance exchange occurs, while <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha \rightarrow +\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> corresponds to a situation that can be interpreted as the cells forming a single structure. We also clarify the relationship between the asymptotic analysis with respect to the parameter <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and the asymptotics of the system in connection with the convergence of convex functionals known as Mosco convergence. Finally, we consider time-dependent permeability and analyze the case where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> 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.</p>

Asymptotic analysis of transmission problems with parameter-dependent Robin conditions

期刊论文
发表日期：

2026年6月20日

查看全文

Takeshi Fukao

摘要

We study a transmission problem of Neumann–Robin type involving a parameter $\alpha $ $α$ and perform an asymptotic analysis with respect to $\alpha $ $α$ . The limits $\alpha \rightarrow 0$ $α \to 0$ and $\alpha \rightarrow +\infty $ $α \to + \infty$ 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 $ $α$ : the case $\alpha \rightarrow 0$ $α \to 0$ 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 \rightarrow +\infty $ $α \to + \infty$ 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 $ $α$ and the asymptotics of the system in connection with the convergence of convex functionals known as Mosco convergence. Finally, we consider time-dependent permeability and analyze the case where $\alpha $ $α$ 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.