Well-posedness and fast-diffusion limit for a bulk–surface reaction–diffusion system

Turing Patterning in Stratified Domains

“…A second class of model considers bulk-surface coupling, where one component is confined to the boundary of the main bulk domain, and reactants flow between the two regions, such as in the case of proteins diffusing in the cytoplasm and binding on the cell membrane (Rätz and Röger 2014;Madzvamuse et al 2015;Spill et al 2016;Cusseddu et al 2018;Paquin-Lefebvre et al 2018;Frey et al 2018). There is substantial recent interest in such models, from very theoretical results on existence and fast reaction-limiting behavior (Rätz 2015;Anguige and Röger 2017;Hausberg and Röger 2018), to spike dynamics (Gomez et al 2018) and a myriad of applications to understanding cell polarity (Thalmeier et al 2016;Kretschmer and Schwille 2016;Geßele et al 2020). One particularly well studied example is the pole-to-pole Min protein oscillation in E. coli, which has the biological function of guiding the cell division machinery to midcell (Kretschmer and Schwille 2016).…”

“… 2018 ). There is substantial recent interest in such models, from very theoretical results on existence and fast reaction-limiting behavior (Rätz 2015 ; Anguige and Röger 2017 ; Hausberg and Röger 2018 ), to spike dynamics (Gomez et al. 2018 ) and a myriad of applications to understanding cell polarity (Thalmeier et al.…”

Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type

“…Such estimates are also available for the corresponding operators on compact manifolds, see [33,Chapter 15, Section 1, (1. 15)].…”

“…For a twovariable bulk-surface system with specific nonlinearities satisfying a monotonicity condition, the well-posedness and exponential convergence to equilibrium is shown in [10] by an entropy method. For a bulk-surface reaction-drift-diffusion system well-posedness and an asymptotic reduction to a non-local reaction-drift-diffusion system on the membrane in the fast-diffusion limit has been proved in [15]. Results on global well-posedness and exponential stability of equilibria have been shown in [8] for transmission type systems with a bulk-interface interaction that represents a general gradient structure.…”

“…We will use references to standard regularity results for parabolic problems on open subsets of R n also for the case in which the spatial domain is a submanifold. This can be justified by a partition of unity subordinate to parametrized subsets of the manifold, see for example [15].…”