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

Abstract

<p style='text-indent:20px;'>We consider a Gierer-Meinhardt system on a surface coupled with a parabolic PDE in the bulk, the domain confined by this surface. Such a model was recently proposed and analyzed for two-dimensional bulk domains by Gomez, Ward and Wei (<i>SIAM J. Appl. Dyn. Syst. 18</i>, 2019). We prove the well-posedness of the bulk-surface system in arbitrary space dimensions and show that solutions remain uniformly bounded in parabolic Hölder spaces for all times. The cytosolic diffusion is typically much larger than the lateral diffusion on the membrane. This motivates to a corresponding asymptotic reduction, which consists of a nonlocal system on the membrane. We prove the convergence of solutions of the full system towards unique solutions of the reduction.</p>