Exponential stability for second order evolutionary problems

Trostorff, Sascha

doi:10.1016/j.jmaa.2015.04.046

Cited by 7 publications

(8 citation statements)

References 18 publications

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“…Thus, as ϕ is positive and compactly supported, ψ(t) = t −∞ ϕ(τ )dτ is eventually constant. Hence, the limit equation is not exponentially stable in the sense of [5, Definition 3.1] (see also [7,Section 3.1]).…”

Section: The Hyperbolic-parabolic Systemmentioning

confidence: 99%

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Stabilization via homogenization

Waurick

2016

Applied Mathematics Letters

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In this short note we treat a 1+1-dimensional system of changing type. On different spatial domains the system is of hyperbolic and elliptic type, that is, formally, ∂ 2 t u n − ∂ 2 x u n = ∂ t f and u n − ∂ 2 x u n = f on the respective spatial domains j∈{1,...,n} j−1 n , 2j−1 2n and j∈{1,...,n} 2j−1 2n , j n . We show that (u n ) n converges weakly to u, which solves the exponentially stable limit equationIf the elliptic equation is replaced by a parabolic one, the limit equation is not exponentially stable.

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Section: The Hyperbolic-parabolic Systemmentioning

confidence: 99%

“…subject to homogeneous Neumann boundary conditions. Moreover, we find that the limit equation is not exponentially stable (in the sense of [5, Definition 3.1], see also [7,Section 3.1]). For the proof of the homogenization (i.e.…”

Section: Introductionmentioning

confidence: 99%

Stabilization via homogenization

Waurick

2016

Applied Mathematics Letters

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“…Having this notion at hand we are able to introduce evolutionary equations and to define well-posedness and exponential stability within this framework. We follow the definition given in [3].…”

Section: Evolutionary Problems and Exponential Stabilitymentioning

confidence: 99%

“…We are now able to state the stability result for (1). Theorem 3.3 [3,Theorem 4.3] Let Ω ⊆ R n such that Poincaré's inequality holds (e.g. Ω bounded).…”

Section: Second-order Problemsmentioning

confidence: 99%

Exponential stability of a second‐order integro‐differential equation with delay

Trostorff

2015

Proc Appl Math and Mech

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“…More precisely, we ask whether the solution decays to zero with an exponential rate, provided that the given right-hand side f does. This property is known as exponential stability and was studied in [Tro13b,Tro14b,Tro15a]. The first difficulty is that the solutions of evolutionary problems are not continuous, so a classical point-wise estimate cannot be used.…”

Section: Introductionmentioning

confidence: 99%