Non-Autonomous Forms and Invariance

We generalize the Beurling--Deny--Ouhabaz criterion for parabolic evolution equations governed by forms to the non-autonomous, non-homogeneous and semilinear case. Let $V, H$ are Hilbert spaces such that $V$ is continuously and densely embedded in $H$ and let $\mathcal{A}(t)\colon V\to V^\prime$ be the operator associated with a bounded $H$-elliptic form $\mathfrak{a}(t,.,.)\colon V\times V \to \mathbb{C}$ for all $t \in [0,T]$. Suppose $\mathcal{C} \subset H$ is closed and convex and $P \colon H \to H$ the orthogonal projection onto $\mathcal{C}$. Given $f \in L^2(0,T;V')$ and $u_0\in \mathcal{C}$, we investigate whenever the solution of the non-autonomous evolutionary problem $$u'(t)+\mathcal{A}(t)u(t)=f(t), \quad u(0)=u_0,$$ remains in $\mathcal{C}$ and show that this is the case if Pu(t) \in V \quad \text{and} \quad \operatorname{Re} \mathfrak{a}(t,Pu(t),u(t)-Pu(t)) \ge \operatorname{Re} \langle f(t), u(t)-Pu(t) \rangle for a.e.\ $t \in [0,T]$. Moreover, we examine necessity of this condition and apply this result to a semilinear problem.