Applied Mathematics & Optimization

, Volume 70, Issue 3, pp 565–566 | Cite as

Erratum to: Decay Rates to Equilibrium for Nonlinear Plate Equations with Degenerate, Geometrically-Constrained Damping

Erratum

1 Erratum to: Appl Math Optim (2013) 68: 361–390DOI 10.1007/s00245-013-9210-8

It was brought to our attention that line 15 on page 374—which claims \(u_t =0\) in \(\Omega \times \Theta \)—is unclear. Upon further inspection, this statement should differentiate between two cases: (a) \(p({\mathbf {x}})= 0\) on \(\Omega \) and (b) \(p({\mathbf {x}}) \ne 0\) (taken in the \(L_2(\Omega )\) sense). Case (a): As argued in the treatment, the application of Holmgren’s theorem gives \(u= 0\) on \(\omega \times \Theta \). The desired final conclusion follows by inserting this information into Berger’s equation (by assumption, no longer containing the damping term \(d({\mathbf {x}})g(u_t)\)), and then employing Kim’s unique continuation result Theorem 3.1. Case (b): One reaches a contradiction with the additional mild assumption that there exists a set of positive measure \(U \subset \omega \) so that \(p({\mathbf {x}})\ne 0\) on \(U\). Thus Theorem 3.2, and consequently Theorems 2.5, 2.6 (Main Results), remain valid when (a) \( p({\mathbf {x}}) \equiv 0\), and when (b) \(p({\mathbf {x}}) \in L_2(\Omega )\) is non-trivial, assuming that \(p({\mathbf {x}}) \ne 0\) on some open set \(U\subset \omega \).

Remark 0.1

If \(p({\mathbf {x}})\) has sufficient regularity then, via the bootstrapping procedure in the proof of Lemma 3.4, the equality
$$\begin{aligned} u_{tt}+\Delta ^2u=f_B(u)+p({\mathbf {x}}) \end{aligned}$$
holds pointwisely; by inserting \(u\equiv 0\) on \(\omega \times \Theta \) we may reach a contradiction to case (b) (as above) if there is a single point \({\mathbf {x}}_0 \subset \omega \) such that \(p({\mathbf {x}}_0) \ne 0\).

We also note that Kim’s theorem is more robust: p(\(\mathbf {x}\)) may be replaced by \(p(u)\) (under suitable dissipativity assumptions); in this case the above proof of the unique continuation result will remain valid.

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Hacettepe UniversityAnkaraTurkey
  2. 2.Oregon State UniversityCorvallisUSA

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