Applied Mathematics & Optimization

, Volume 70, Issue 3, pp 565–566

# 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.