Maximal Regularity for Evolution Equations Governed by Non-Autonomous Forms
Language
en
Article de revue
This item was published in
Advances in Differential Equations. 2014p. ???
Khayyam Publishing
English Abstract
\begin{abstract}\label{abstract} We consider a non-autonomous evolutionary problem \[ \dot{u} (t)+\A(t)u(t)=f(t), \quad u(0)=u_0 \] where the operator $\A(t):V\to V^\prime$ is associated with a form $\fra(t,.,.):V\times V ...Read more >
\begin{abstract}\label{abstract} We consider a non-autonomous evolutionary problem \[ \dot{u} (t)+\A(t)u(t)=f(t), \quad u(0)=u_0 \] where the operator $\A(t):V\to V^\prime$ is associated with a form $\fra(t,.,.):V\times V \to \R$ and $u_0\in V$. Our main concern is to prove well-posedness with maximal regularity which means the following. Given a Hilbert space $H$ such that $V$ is continuously and densely embedded into $H$ and given $f\in L^2(0,T;H)$ we are interested in solutions $u \in H^1(0,T;H)\cap L^2(0,T;V)$. We do prove well-posedness in this sense whenever the form is piecewise Lipschitz-continuous and symmetric. Moreover, we show that each solution is in $C([0,T];V)$. We apply the results to non-autonomous Robin-boundary conditions and also use maximal regularity to solve a quasilinear problem.Read less <
English Keywords
Sesquilinear forms
non-autonomous evolution equations
maximal regularity
non-linear heat equations
ANR Project
Aux frontières de l'analyse Harmonique - ANR-12-BS01-0013
Origin
Hal imported