On a Theorem by W. von Wahl

Abstract

Let $A$ be the - not necessarily densely defined - generator of an analytic semigroup acting in some Banach space $X$. In the paper we prove a general theorem about the existence and uniqueness of solutions of

Our main assumption with respect to the non-linearity is that $F$ is locally Lipschitz continuous with respect to certain intermediate spaces between $\mathcal{D}(A)$ and $X$. Our theorem extends results obtained by W. von Wahl [9] and A. Lunardi [2]. In the second part this theorem is applied to the Cahn-Hilliard equation with Dirichlet boundary conditions.