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We consider a model system consisting of two reaction–diffusion equations, where one species diffuses in a volume while the other species diffuses on the surface which surrounds the volume. The two equations are coupled via a nonlinear reversible Robin-type boundary condition for the volume species and a matching reversible source term for the boundary species. As a consequence of the coupling, the total mass of the two species is conserved. The considered system is motivated for instance by models for asymmetric stem cell division.
Firstly we prove the existence of a unique weak solution via an iterative method of converging upper and lower solutions to overcome the difficulties of the nonlinear boundary terms. Secondly, our main result shows explicit exponential convergence to equilibrium via an entropy method after deriving a suitable entropy entropy-dissipation estimate for the considered nonlinear volume-surface reaction–diffusion system.
Keywords: Volume-surface reaction–diffusion, Nonlinear boundary conditions, Global existence, Exponential convergence to equilibrium
@article{AIHPC_2018__35_3_643_0,
author = {Fellner, Klemens and Latos, Evangelos and Tang, Bao Quoc},
title = {Well-posedness and exponential equilibration of a volume-surface reaction{\textendash}diffusion system with nonlinear boundary coupling},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {643--673},
publisher = {Elsevier},
volume = {35},
number = {3},
year = {2018},
doi = {10.1016/j.anihpc.2017.07.002},
mrnumber = {3778646},
zbl = {1392.35186},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.07.002/}
}
TY - JOUR AU - Fellner, Klemens AU - Latos, Evangelos AU - Tang, Bao Quoc TI - Well-posedness and exponential equilibration of a volume-surface reaction–diffusion system with nonlinear boundary coupling JO - Annales de l'I.H.P. Analyse non linéaire PY - 2018 SP - 643 EP - 673 VL - 35 IS - 3 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.07.002/ DO - 10.1016/j.anihpc.2017.07.002 LA - en ID - AIHPC_2018__35_3_643_0 ER -
%0 Journal Article %A Fellner, Klemens %A Latos, Evangelos %A Tang, Bao Quoc %T Well-posedness and exponential equilibration of a volume-surface reaction–diffusion system with nonlinear boundary coupling %J Annales de l'I.H.P. Analyse non linéaire %D 2018 %P 643-673 %V 35 %N 3 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.07.002/ %R 10.1016/j.anihpc.2017.07.002 %G en %F AIHPC_2018__35_3_643_0
Fellner, Klemens; Latos, Evangelos; Tang, Bao Quoc. Well-posedness and exponential equilibration of a volume-surface reaction–diffusion system with nonlinear boundary coupling. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 643-673. doi: 10.1016/j.anihpc.2017.07.002
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