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The rate at which energy decays in a damped string. (English) Zbl 0818.35072
The authors treat the initial boundary value problem \[ u_{tt} - u_{xx} + 2a(x)u_ t = 0,\;0 < x < 1,\;0 < t; \]
\[ u(0,t) = u(1,t) = 0,\;u(x,0) = u_ 0(x),\;u_ t(x,0) = v_ 0(x). \] Here \(a(x) \in L^ 2 (0,1)\), \(0 \leq \alpha \leq a(x) \leq \beta < \infty\). The decay rate is defined as \[ \omega (a) = \inf \{\omega;\;\exists C (\omega) > 0, \quad \text{such that} \quad E(t) \equiv \int^ 1_ 0 (u^ 2_ x + u^ 2_ t)dx \leq CE(0) e^{2 \omega t}\}. \] The equation is interpreted as the system \(V_ t = AV\); \(V = [u,u_ t]\), \[ A = \left( \begin{matrix} 0 & I \\ d^ 2/dx^ 2 & - 2a \end{matrix} \right) : D(A) \to X = H^ 1_ 0 (0,1) \times L^ 2 (0,1). \] The spectral abscissa of \(A\) is \(\mu (a) = \sup \{\text{Re} \lambda; \lambda \in \sigma (A)\}\). The authors establish necessary and sufficient conditions for the presence of real eigenvalues of \(A\) and give affirmative reply to a conjecture by J. Rauch [Arch. Ration Mech. Anal. 62, 77-85 (1976; Zbl 0335.35062)]. Furthermore, by using their Riesz basis for \(X\) they prove the equality \(\mu (a) = \omega (a)\) under the condition such that \(a\) is of bounded variation.
Reviewer: H.Yamagata (Osaka)

MSC:
35P15 Estimates of eigenvalues in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
47A10 Spectrum, resolvent
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