  In this paper, we provide well-posedness results for nonlinear parabolic PDEs
given by reaction-diffusion equations describing the concentration of oxygen in
encapsulated cells. The cells are described in terms of a core and a shell,
which introduces a discontinuous diffusion coefficient as the material
properties of the core and shell differ. In addition, the cells are subject to
general nonlinear consumption of oxygen. As no monotonicity condition is
imposed on the consumption monotone operator theory cannot be used. Moreover,
the discontinuity in the diffusion coefficient bars us to apply classical
results. However, by directly applying a Galerkin method we obtain uniqueness
and existence of the strong form solution. These results will provide the basis
to study the dynamics of cells in critical states.
