Well-posedness and exponential equilibration of a volume-surface reaction–diffusion system with nonlinear boundary coupling

Fellner, Klemens; Latos, Evangelos; Tang, Bao Quoc

doi:10.1016/j.anihpc.2017.07.002

Fellner, Klemens ; Latos, Evangelos ; Tang, Bao Quoc

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 643-673.

Résumé

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.

MR Zbl

DOI : 10.1016/j.anihpc.2017.07.002

Classification : 35K61, 35A01, 35B40, 35K57
Mots clés : Volume-surface reaction–diffusion, Nonlinear boundary conditions, Global existence, Exponential convergence to equilibrium

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     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},
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     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2017.07.002/}
}

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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. http://archive.numdam.org/articles/10.1016/j.anihpc.2017.07.002/

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