# On a Theorem by W. von Wahl

### M. Uiterdijk

Delft University of Technology, Netherlands

## 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

$u’(t)=Au(t)+F(u(t))$

$u(0)=u_{0}.$

Our main assumption with respect to the non-linearity is that $F$ is locally Lipschitz continuous with respect to certain intermediate spaces between $D(A)$ 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.

## Cite this article

M. Uiterdijk, On a Theorem by W. von Wahl. Z. Anal. Anwend. 16 (1997), no. 4, pp. 961–978

DOI 10.4171/ZAA/799