Invariance of Convex Sets for Non-autonomous Evolution Equations Governed by Forms

We consider a non-autonomous form $\text{fra}:[0,T]\times V\times V\to \text{C}$ where $V$ is a Hilbert space which is densely and continuously embedded in another Hilbert space $H$. Denote by $\text{A}(t)\in \text{L}(V,V^{\mathrm{\prime }})$ the associated operator. Given $f\in L^2(0,T,V^{\mathrm{\prime }})$, one knows that for each $u_0\in H$ there is a unique solution $u\in H^1(0,T;V^{\mathrm{\prime }})\cap L^2(0,T;V)$ of $$\dot{u}(t)+\text{A}(t)u(t)=f(t),u(0)=u_0\mathrm{.}$$ %\begin{align*} %&\dot u(t) + \A(t)u(t)= f(t)\ %& u(0)=u_0. %\end{align*} This result by J. L. Lions is well-known. The aim of this article is to find a criterion for the invariance of a closed convex subset $\text{Conv}$ of $H$; i.e.\ we give a criterion on the form which implies that $u(t)\in \text{Conv}$ for all $t\in [0,T]$ whenever $u_0\in \text{Conv}$. In the autonomous case for $f=0$, the criterion is known and even equivalent to invariance by a result proved in \cite{Ouh96} (see also \cite{Ouh05}). We give applications to positivity and comparison of solutions to heat equations with non-autonomous Robin boundary conditions. We also prove positivity of the solution to a quasi-linear heat equation.