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

Wolfgang Arendt (1) , Dominik Dier (1) , El Maati Ouhabaz (2)

1 Institut for Applied Analysis
2 IMB - Institut de Mathématiques de Bordeaux

We consider a non-autonomous form $\fra:[0,T]\times V\times V \to \C$ where $V$ is a Hilbert space which is densely and continuously embedded in another Hilbert space $H$. Denote by $\A(t) \in \L(V,V')$ the associated operator. Given $f \in L^2(0,T, V')$, one knows that for each $u_0 \in H$ there is a unique solution $u\in H^1(0,T;V')\cap L^2(0,T;V)$ of $$\dot u(t) + \A(t) u(t) = f(t), \, \, u(0) = u_0.$$ %\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 $\Conv$ of $H$; i.e.\ we give a criterion on the form which implies that $u(t)\in \Conv$ for all $t\in[0,T]$ whenever $u_0\in\Conv$. 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.

Mots clés: Sesquilinear forms; non-autonomous evolution equations; non-linear heat equations; invariance of closed convex sets

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Wolfgang Arendt, Dominik Dier, El Maati Ouhabaz. Invariance of Convex Sets for Non-autonomous Evolution Equations Governed by Forms. J. London Math. Soc., 2014, pp.14. ⟨10.1112/jlms/jdt082⟩. ⟨hal-00797182⟩