ARENDT, Wolfgang; DIER, Dominik; OUHABAZ, El Maati

doi:10.1112/jlms/jdt082

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ARENDT, Wolfgang
Institut for Applied Analysis

DIER, Dominik
Institut for Applied Analysis

OUHABAZ, El Maati
Institut de Mathématiques de Bordeaux [IMB]

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Este ítem está publicado en

J. London Math. Soc. 2014p. 14

Resumen en inglés

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.< Leer menos

Palabras clave en inglés

Sesquilinear forms

non-autonomous evolution equations

non-linear heat equations

invariance of closed convex sets

URI

https://oskar-bordeaux.fr/handle/20.500.12278/189656

DOI

http://dx.doi.org/10.1112/jlms/jdt082

Proyecto ANR

Aux frontières de l'analyse Harmonique - ANR-12-BS01-0013

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Importado de Hal

Centros de investigación

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251