  We study the global existence of classical solutions to volume-surface
reaction-diffusion systems with control of mass. Such systems appear naturally
from modeling evolution of concentrations or densities appearing both in a
volume domain and its surface, and therefore have attracted considerable
attention. Due to the characteristic volume-surface coupling, global existence
of solutions to general systems is a challenging issue. In particular, the
duality method, which is powerful in dealing with mass conserved systems in
domains, is not applicable on its own. In this paper, we introduce a new family
of $L^p$-energy functions and combine them with a suitable duality method for
volume-surface systems, to ultimately obtain global existence of classical
solutions under a general assumption called the \textit{intermediate sum
condition}. For systems that conserve mass, but do not satisfy this condition,
global solutions are shown under a quasi-uniform condition, that is, under the
assumption that the diffusion coefficients are close to each other. In the case
of mass dissipation, we also show that the solution is bounded uniformly in
time by studying the system on each time-space cylinder of unit size, and
showing that the solution is sup-norm bounded independently of the cylinder.
Applications of our results include global existence and boundedness for
systems arising from membrane protein clustering or activation of Cdc42 in cell
polarization.
