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

  • Klemens Fellner

    Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria
  • Evangelos Latos

    Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria
  • Bao Quoc Tang

    Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria

Abstract

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.

Cite this article

Klemens Fellner, Evangelos Latos, Bao Quoc Tang, Well-posedness and exponential equilibration of a volume-surface reaction–diffusion system with nonlinear boundary coupling. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 3, pp. 643–673

DOI 10.1016/J.ANIHPC.2017.07.002