This paper considers the existence of local and global-in-time strong
solutions to the advection-diffusion equation with variable coefficients
on an evolving surface with a boundary. We apply both the maximal
-in-time regularity
for Hilbert space-valued functions and semigroup theory to construct local and global-in-time strong
solutions to an evolution equation. Using the approach and our function spaces on the evolving
surface, we show the existence of local and global-in-time strong solutions to the advection-diffusion
equation. Moreover, we derive the asymptotic stability of the global-in-time strong solution.
Keywords
advection-diffusion equation with variable coefficients,
time-dependent Laplace–Beltrami operator, function spaces
on evolving surfaces, maximal $L^p$-regularity, asymptotic
stability
