Bulk-surface systems on evolving domains

Abstract Global existence and large time behaviour for a class of bulk-surface reaction–diffusion systems in evolving domains are studied. Such problems appear typically from modelling receptor–ligand dynamics in biological cells. Our first main result is the global existence and boundedness of solutions in all dimensions. This is achieved by proving $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -maximal regularity of parabolic equations and duality methods in moving surfaces, which are of independent interest. The second main result is the large time dynamics where we show, under the assumption that the volume/area of the moving domain/surface is unchanged and that the material velocities are decaying for large time that the solution converges to a unique spatially homogeneous equilibrium. This is proved by extending the entropy method to bulk-surface systems in evolving domains.