An abstract existence theorem for parabolic systems

In this paper we prove an abstract existence theorem which can be applied to solve parabolic problems in a wide range of applications. It also applies to parabolic variational inequalities. The abstract theorem is based on a Gelfand triple $(V,H,V^*)$, where the standard realization for parabolic systems of second order is $(W^{1, 2}(\Omega),L^2(\Omega), W^{1,2}(\Omega)^*)$. But also realizations to other problems are possible, for example, to fourth order systems. In all applications to boundary value problems the set $M\subset V$ is an affine subspace, whereas for variational inequalities the constraint $M$ is a closed convex set. The proof is purely abstract and new. The corresponding compactness theorem is based on [5]. The present paper is suitable for lectures, since it relays on the corresponding abstract elliptic theory.