Stabilization and asymptotic behavior of a generalized telegraph equation
type: Item
instance of: scholarly article
title: Stabilization and asymptotic behavior of a generalized telegraph equation
zbMATH Open document ID: 1347.35151
author: Serge Nicaise
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2016-01-05T00:00:00Z
review text: The author studies the following generalization of the telegraph equation: \[ \begin{cases} V_t + gV + aI_x + kW = 0 & \text{in } (0,L) \times (0, \infty), \\ I_t + rI + b V_x = 0 & \text{in } (0,L) \times (0,\infty), \\ W_t + cW = V & \text{in } \Omega \times (0, \infty), \\ V(0,\cdot) = V(L,\cdot) = 0 & \text{on } (0, \infty), \\ V(\cdot,0) = V_0, I(\cdot,0) = I_0, W(\cdot,0) = W_0, & \text{in } (0,L) \end{cases} \] for some \(L^\infty\) coefficients which are additionally assumed to satisfy \(a \geq \delta\), \(b \geq \delta\), \(c \geq \delta\) and \(k+g \geq \delta\) for some \(\delta > 0\). It is shown that for a suitable energy norm the realization of the above problem generates a \(C_0\)-semigroup. Further, the strong stability of the semigroup is obtained with the help of the ABVL theorem. Afterwards, the decay rates of the solutions are investigated in dependence of the coefficients. The author establishes polynomial decay and under some additional assumptions even exponential decay of the solutions. In the special case of constant coefficients the optimality of the decay rates is shown.
reviewed by: Stephan Fackler
Mathematics Subject Classification ID: 35L50
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 6526491
zbMATH Keywords: telegraph equation
zbMATH Keywords: stabilization
zbMATH Keywords: \(L^\infty\) coefficients
zbMATH Keywords: \(C_0\)-semigroup
zbMATH Keywords: decay rates
zbMATH Keywords: polynomial decay
zbMATH Keywords: exponential decay
MaRDI profile type: Publication
full work available at URL: s00033-015-0568-0
OpenAlex ID: W4308855473
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DOI: 10.1007/S00033-015-0568-0
description: scientific article; zbMATH DE number 6526491