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David Seifert

Publications and source records attributed to David Seifert.

At least 19 recordsLinked to original sources
arXiv2026

Optimal energy decay rates for Klein-Gordon equations with Kelvin-Voigt damping

We study the long-time behaviour of solutions to a one-dimensional linear Klein-Gordon equation with Kelvin-Voigt damping. One of the interesting features of the equation is that the generator of the associated $C_0$-semigroup has multiple spectral points on the imaginary axis. As our main result, we show that the energy of every possible solution converges to zero as time goes to infinity and, moreover, we provide an optimal polynomial energy decay rate for a certain class of solutions.

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arXiv2025

Polynomial Stability of Non-Linearly Damped Contraction Semigroups

We investigate the stability properties of an abstract class of semi-linear systems. Our main result establishes rational rates of decay for classical solutions assuming a certain non-uniform observability estimate for the linear part and suitable conditions on the non-linearity. We illustrate the strength of our abstract results by applying them to a one-dimensional wave equation with weak non-linear damping and to an Euler-Bernoulli beam with a tip mass subject to non-linear damping.

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arXiv2025

Tauberian theorems for sequences and the Katznelson--Tzafriri theorem

In this note, we present an alternative proof of a quantified Tauberian theorem for vector-valued sequences first proved in \cite{Sei15_Tauberian}. The theorem relates the decay rate of a bounded sequence with properties of a certain boundary function. We present a slightly strengthened version of this result, and illustrate how it can be used to obtain quantified versions of the Katznelson--Tzafriri theorem as well as results on Ritt operators.

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arXiv2024

Admissibility theory in abstract Sobolev scales and transfer function growth at high frequencies

For strongly continous semigroups on Hilbert spaces, we investigate admissibility properties of control and observation operators shifted along continuous scales of spaces built by means of either interpolation and extrapolation or functional calculus. Our results show equivalence of admissibility in, on the one hand, a fractional domain of the generator and, on the other hand, a (different, in general) quadratic interpolation space of the same "Sobolev order". Furthermore, such properties imply quantified resolvent bounds in the original state space topology. When the semigroup is a group, the resulting frequency-domain estimates are in fact equivalent to the aforementioned time-domain properties. In the case of systems with both control and observation, we are able to translate input-output regularity properties into high-frequency growth rates of operator-valued transfer functions. As an application, based on results by Lasiecka, Triggiani and Tataru on interior and boundary regularity of the wave equation under Neumann control, we derive optimal asymptotics for the Neumann-to-Dirichlet wave transfer function. With that in hand, we establish non-uniform energy decay rates for the wave equation posed in a rectangle and subject to Neumann damping on an arbitrary open subset of the boundary.

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arXiv2024

A non-uniform Datko-Pazy theorem for bounded operator semigroups

We present a non-uniform analogue of the classical Datko-Pazy theorem. Our main result shows that an integrability condition imposed on orbits originating in a fractional domain of the generator (as opposed to all orbits) implies polynomial stability of a bounded $C_0$-semigroup. As an application of this result we establish polynomial stability of a semigroup under a certain non-uniform Lyapunov-type condition. We moreover give a new proof, under slightly weaker assumptions, of a recent result deducing polynomial stability from a certain non-uniform observability condition.

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arXiv2024

Polynomial stability of a coupled wave-heat network

We study the long-time asymptotic behaviour of a topologically non-trivial network of wave and heat equations. By analysing the simpler wave and the heat networks separately, and then applying recent results for abstract coupled systems, we establish energy decay at the rate $t^{-4}$ as $t\to\infty$ for all classical solutions.

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arXiv2024

The asymptotic behaviour of the Ces\`aro operator

We study the asymptotic behaviour of orbits $(T^nx)_{n\ge0}$ of the classical Ces\`aro operator $T$ for sequences $x$ in the Banach space $c$ of convergent sequences. We give new non-probabilistic proofs, based on the Katznelson-Tzafriri theorem and one of its quantified variants, of results which characterise the set of sequences $x\in c$ that lead to convergent orbits and, for sequences satisfying a simple additional condition, we provide a rate of convergence. These results are then shown, again by operator-theoretic techniques, to be optimal in different ways. Finally, we study the asymptotic behaviour of the Ces\`aro operator defined on spaces of continuous functions, establishing new and improved results in this setting, too.

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arXiv2024

Stability of abstract coupled systems

We study stability of abstract differential equations coupled by means of a general algebraic condition. Our approach is based on techniques from operator theory and systems theory, and it allows us to study coupled systems by exploiting properties of the components, which are typically much simpler to analyse. As our main results we establish resolvent estimates and decay rates for abstract boundary-coupled systems. We illustrate the power of the general results by using them to obtain rates of energy decay in coupled systems of one-dimensional wave and heat equations, and in a wave equation with an acoustic boundary condition.

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arXiv2023

Non-uniform Stability of Damped Contraction Semigroups

We investigate the stability properties of strongly continuous semigroups generated by operators of the form $A-BB^\ast$, where $A$ is a generator of a contraction semigroup and $B$ is a possibly unbounded operator. Such systems arise naturally in the study of hyperbolic partial differential equations with damping on the boundary or inside the spatial domain. As our main results we present general sufficient conditions for non-uniform stability of the semigroup generated by $A-BB^\ast$ in terms of selected observability-type conditions of the pair $(B^\ast,A)$. We apply the abstract results to obtain rates of energy decay in one-dimensional and two-dimensional wave equations, a damped fractional Klein--Gordon equation and a weakly damped beam equation.

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arXiv2022

Optimal decay for a wave-heat system with Coleman-Gurtin thermal law

We study the long-term behaviour of solutions to a one-dimensional coupled wave-heat system with Coleman-Gurtin thermal law. Our approach is based on the asymptotic theory of $C_0$-semigroups and recent results developed for coupled control systems. As our main results, we represent the system as a feedback interconnection between the wave part and the Coleman-Gurtin part and we show that the associated semigroup in the history framework of Dafermos is polynomially stable with optimal decay rate $t^{-2}$ as $t\to\infty$. In particular, we obtain a sharp estimate for the rate of energy decay of classical solutions to the problem.

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arXiv2022

Some developments around the Katznelson-Tzafriri theorem

This paper is a survey article on developments arising from a theorem proved by Katznelson and Tzafriri in 1986 showing that $\lim_{n\to\infty} \|T^n(I-T)\| =0$ if $T$ is a power-bounded operator on a Banach space and $\sigma(T) \cap \T \subseteq \{1\}$. Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory.

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arXiv2022

A Katznelson-Tzafriri theorem for analytic Besov functions of operators

Let $T$ be a power-bounded operator on a Banach space $X$, $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the unit disc $\mathbb{D}$, and assume that there is a bounded functional calculus for the operator $T$, so there is a bounded algebra homomorphism mapping functions $f \in \mathcal{A}$ to bounded operators $f(T)$ on $X$. Theorems of Katznelson-Tzafriri type establish that $\lim_{n\to\infty} \|T^n f(T)\| = 0$ for functions $f \in \mathcal{A}$ whose boundary functions vanish on the unitary spectrum $\sigma(T)\cap \mathbb{T}$ of $T$, or sometimes satisfy a stronger assumption of spectral synthesis. We consider the case when $\mathcal{A}$ is the Banach algebra $\mathcal{B}(\mathbb{D})$ of analytic Besov functions on $\mathbb{D}$. We prove a Katznelson-Tzafriri theorem for the $\mathcal{B}(\mathbb{D})$-calculus which extends several previous results.

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arXiv2020

Optimal rates of decay in the Katznelson-Tzafriri theorem for operators on Hilbert spaces

The Katznelson-Tzafriri theorem is a central result in the asymptotic theory of discrete operator semigroups. It states that for a power-bounded operator $T$ on a Banach space we have $||T^n(I-T)\|\to0$ if and only if $σ(T)\cap\mathbb{T}\subseteq\{1\}$. The main result of the present paper gives a sharp estimate for the rate at which this decay occurs for operators on Hilbert space, assuming the growth of the resolvent norms $\|R(e^{iθ},T)\|$ as $|θ|\to0$ satisfies a mild regularity condition. This significantly extends an earlier result by the second author, which covered the important case of polynomial resolvent growth. We further show that, under a natural additional assumption, our condition on the resolvent growth is not only sufficient but also necessary for the conclusion of our main result to hold. By considering a suitable class of Toeplitz operators we show that our theory has natural applications even beyond the setting of normal operators, for which we in addition obtain a more general result.

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arXiv2020

Asymptotics and approximation of large systems of ordinary differential equations

In this paper we continue our earlier investigations into the asymptotic behaviour of infinite systems of coupled differential equations. Under the mild assumption that the so-called characteristic function of our system is completely monotonic we obtain a drastically simplified condition which ensures boundedness of the associates semigroup. If the characteristic function satisfies certain additional conditions we deduce sharp rates of convergence to equilibrium. We moreover address the important and delicate issue of the role of the infinite system in understanding the asymptotic behaviour of large but finite systems, and we provide a precise way of obtaining size-independent rates of convergence for families of finite-dimensional systems. Finally, we illustrate our abstract results in the setting of the well-known platoon problem.

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arXiv2019

Optimal energy decay in a one-dimensional wave-heat system with infinite heat part

Using recent results in the theory of $C_0$-semigroups due to Batty, Chill and Tomilov (J. Eur. Math. Soc. 18(4):853-929, 2016) we study energy decay in a one-dimensional coupled wave-heat system with finite wave part and infinite heat part. Our main result provides a sharp estimate for the rate of energy decay of a certain class of classical solutions. The present paper can be thought of as a natural sequel to a recent work by Batty, Paunonen and Seifert (J. Evol. Equ. 16:649-664, 2016), which studied a similar wave-heat system with finite wave and heat parts using a celebrated result due to Borichev and Tomilov.

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arXiv2019

Optimality of the quantified Ingham-Karamata theorem for operator semigroups with general resolvent growth

We prove that a general version of the quantified Ingham-Karamata theorem for $C_0$-semigroups is sharp under mild conditions on the resolvent growth, thus generalising the results contained in a recent paper by the same authors. It follows in particular that the well-known Batty-Duyckaerts theorem is optimal even for bounded $C_0$-semigroups whose generator has subpolynomial resolvent growth. Our proof is based on an elegant application of the open mapping theorem, which we complement by a crucial technical lemma allowing us to strengthen our earlier results.

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