The aim this thesis is to develop several useful tools for the treatment of elliptic and parabolic partial differential equations which arise in the context of certain lithium-ion battery models, for instance. We focus on the setting of one parabolic and two elliptic equations in a bounded domain which are coupled by a nonlinearity on the right-hand side. This nonlinear function is assumed to possess a monotonicity property and may show exponential growth as well as singular behavior with respect to the unknown functions.In order to make fixed point theorems of Schauder type applicable to the linearized equations, results from the Lp-theory of parameter-elliptic systems are generalized to function spaces of higher regularity, yielding the compactness of the corresponding solution operator. Using reflection techniques, our results are transferred to the case of a rectangular geometry of the domain for a certain class of parameter-elliptic boundary value problems.Furthermore, discontinuities of the coefficients in the equations across internal interfaces are treated by the consideration of transmission problems. In particular, we prove the property of maximal regularity for the realization of general parabolic transmission problems.Our main result states the local in time existence of a strong solution to an elliptic-parabolic system of the indicated structure
