On the right multiplicative perturbation of non-autonomous $L^p$-maximal regularity

Augner, Björn; Jacob, Birgit; Laasri, Hafida

doi:10.7900/jot.2014jul31.2064

Cited by 8 publications

(18 citation statements)

References 27 publications

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“…The two problems are motivated by applications to semi-linear evolution equations and boundary value problems. We extend the results in [1] and [3] in three directions. The first one is to consider general forms which may not satisfy the Kato square root property, a condition which was used in an essential way in the previous two papers.…”

Section: Introductionsupporting

confidence: 66%

“…where B(t) and P (t) are bounded operators on H such that Re (B(t) −1 g, g) ≥ δ g 2 for some δ > 0 and all g ∈ H. The left perturbation problem (1.4) was already considered by Arendt et al [1] and the right perturbation one (1.5) by Augner et al [3]. The two problems are motivated by applications to semi-linear evolution equations and boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

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Non-autonomous right and left multiplicative perturbations and maximal regularity

Achache

Ouhabaz

2018

Studia Math.

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We consider the problem of maximal regularity for non-autonomous Cauchy problemsIn both cases, the time dependent operators A(t) are associated with a family of sesquilinear forms and the multiplicative left or right perturbations B(t) as well as the additive perturbation P (t) are families of bounded operators on the considered Hilbert space. We prove maximal L p -regularity results and other regularity properties for the solutions of the previous problems under minimal regularity assumptions on the forms and perturbations.keywords: Maximal regularity, non-autonomous evolution equations, multiplicative and additive perturbations. Mathematics Subject Classification (2010): 35K90, 35K45, 47D06.

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Section: Introductionsupporting

confidence: 66%

Section: Introductionmentioning

confidence: 99%

Non-autonomous right and left multiplicative perturbations and maximal regularity

Achache

Ouhabaz

2018

Studia Math.

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“…. Then there exist constants τ > 0 and C τ > 0 such that for each classical solution x of (5) we have…”

Section: Non-autonomous Port-hamiltonian Systemsmentioning

confidence: 99%

“…Here, we have used that P 1 is invertible and self adjoint, x solves (5) and that H is self-adjoint as well. Next, by the fundamental theorem of calculus we have…”

Section: Non-autonomous Port-hamiltonian Systemsmentioning

confidence: 99%

Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems

Augner

Jacob

2014

Evolution Equations &Amp; Control Theory

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We study the non-autonomous version of an infinite-dimensional linear port-Hamiltonian system on an interval [a, b]. Employing abstract results on evolution families, we show C 1 -well-posedness of the corresponding Cauchy problem, and thereby existence and uniqueness of classical solutions for sufficiently regular initial data. Further, we demonstrate that a dissipation condition in the style of the dissipation condition sufficient for uniform exponential stability in the autonomous case also leads to a uniform exponential decay rate of the energy in this non-autonomous setting.

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“…More recently, this problem has been studied with some progress and different approaches [4,5,18,11,16,19,13,12,14]. Results on multiplicative perturbation are established in [4,11,6]. See also the recent review paper [3] for more details and references.…”

Section: Introductionmentioning

confidence: 99%

On evolution equations governed by non-autonomous forms

El-Mennaoui

Laasri

2016

Arch. Math.

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We consider a linear non-autonomous evolutionary Cauchy probleṁ1) where the operator A(t) arises from a time depending sesquilinear form a(t, ., .) on a Hilbert space H with constant domain V . Recently, a result on L 2 -maximal regularity in H, i.e. for each given f ∈ L 2 (0, T, H) and u0 ∈ V the problem (0.1) has a unique solution u ∈ L 2 (0, T, V ) ∩ H 1 (0, T, H), is proved in Dier (J. Math. Anal. Appl. 425:33-54, 2015) under the assumption that a is symmetric and of bounded variation. The aim of this paper is to prove that the solutions of an approximate nonautonomous Cauchy problem in which a is symmetric and piecewise affine are closed to the solutions of that governed by symmetric and of bounded variation form. In particular, this provides an alternative proof of the result in Dier (J. Math. Anal. Appl. 425:33-54, 2015) on L 2 -maximal regularity in H. Mathematics Subject Classification. 35K90, 35K50, 35K45, 47D06.

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On the right multiplicative perturbation of non-autonomous $L^p$-maximal regularity

Cited by 8 publications

References 27 publications

Non-autonomous right and left multiplicative perturbations and maximal regularity

Non-autonomous right and left multiplicative perturbations and maximal regularity

Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems

On evolution equations governed by non-autonomous forms

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