Zeitschrift für Analysis und ihre Anwendungen
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Published online: 1997-12-31
On a Theorem by W. von WahlM. Uiterdijk (1) Delft University of Technology, Netherlands
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 $\mathcal D(A)$ and $X$. Our theorem extends results obtained by W. von Wahl  and A. Lunardi . In the second part this theorem is applied to the Cahn-Hilliard equation with Dirichlet boundary conditions.
Keywords: Abstract semilinear parabolic equations, differential operators with non-dense domain, intermediate spaces, Cahn-Hilliard equation
Uiterdijk M.: On a Theorem by W. von Wahl. Z. Anal. Anwend. 16 (1997), 961-978. doi: 10.4171/ZAA/799