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Invariance of Convex Sets for Non-autonomous Evolution Equations Governed by Forms
| hal.structure.identifier | Institut for Applied Analysis | |
| dc.contributor.author | ARENDT, Wolfgang | |
| hal.structure.identifier | Institut for Applied Analysis | |
| dc.contributor.author | DIER, Dominik | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | OUHABAZ, El Maati | |
| dc.date.accessioned | 2024-04-04T02:22:30Z | |
| dc.date.available | 2024-04-04T02:22:30Z | |
| dc.date.created | 2013-02-27 | |
| dc.date.issued | 2014 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/189656 | |
| dc.description.abstractEn | 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. | |
| dc.description.sponsorship | Aux frontières de l'analyse Harmonique - ANR-12-BS01-0013 | |
| dc.language.iso | en | |
| dc.subject.en | Sesquilinear forms | |
| dc.subject.en | non-autonomous evolution equations | |
| dc.subject.en | non-linear heat equations | |
| dc.subject.en | invariance of closed convex sets | |
| dc.title.en | Invariance of Convex Sets for Non-autonomous Evolution Equations Governed by Forms | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.1112/jlms/jdt082 | |
| dc.subject.hal | Mathématiques [math]/Equations aux dérivées partielles [math.AP] | |
| dc.subject.hal | Mathématiques [math]/Analyse fonctionnelle [math.FA] | |
| dc.identifier.arxiv | 1303.1167 | |
| bordeaux.journal | J. London Math. Soc | |
| bordeaux.page | 14 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-00797182 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-00797182v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=J.%20London%20Math.%20Soc&rft.date=2014&rft.spage=14&rft.epage=14&rft.au=ARENDT,%20Wolfgang&DIER,%20Dominik&OUHABAZ,%20El%20Maati&rft.genre=article |
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