Advances in Differential Equations

Maximal regularity for evolution equations governed by non-autonomous forms

Wolfgang Arendt, Dominik Dier, Hafida Laasri, and El Maati Ouhabaz

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Abstract

We consider a non-autonomous evolutionary problem \[ \dot{u} (t)+\mathcal A(t)u(t)=f(t), \quad u(0)=u_0 \] where the operator $\mathcal A(t)\colon V\to V^\prime$ is associated with a form $\mathfrak{a}(t,.,.)\colon V\times V \to \mathbb R$ 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 satisfies the square root property. Moreover, we show that each solution is in $C([0,T];V)$. The results are applied to non-autonomous Robin-boundary conditions and maximal regularity is used to solve a quasilinear problem.

Article information

Source
Adv. Differential Equations Volume 19, Number 11/12 (2014), 1043-1066.

Dates
First available in Project Euclid: 18 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.ade/1408367288

Mathematical Reviews number (MathSciNet)
MR3250762

Zentralblatt MATH identifier
1319.35106

Subjects
Primary: 35K90: Abstract parabolic equations 35K50 35K45: Initial value problems for second-order parabolic systems 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

Citation

Arendt, Wolfgang; Dier, Dominik; Laasri, Hafida; Ouhabaz, El Maati. Maximal regularity for evolution equations governed by non-autonomous forms. Adv. Differential Equations 19 (2014), no. 11/12, 1043--1066. https://projecteuclid.org/euclid.ade/1408367288.


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