Self-excited vibrations for damped and delayed higher dimensional wave equations
type: Item
instance of: scholarly article
title: Self-excited vibrations for damped and delayed higher dimensional wave equations
zbMATH Open document ID: 1426.35024
author: Nemanja Kosovalić
author: B. Pigott
published in: Discrete and Continuous Dynamical Systems
publication date: 2019-03-28T00:00:00Z
review text: This paper concerns autonomous dissipative delayed semilinear wave equations on \(d\)-dimensional cubes with homogeneous Dirichlet boundary conditions. The authors construct one-parameter families of time-periodic solutions which bifurcate from a stationary solution. The equations are quite special (because of symmetry assumptions), and the solutions are quite special (because they have a fixed prescribed special frequency vector). The authors mention that ``we will see some surprising features which are not encountered in the one-dimensional problem''. But, in fact, this is not surprising because since the celebrated counterexample of \textit{M. Renardy} [Z. Angew. Math. Phys. 45, No. 6, 854--865 (1994; Zbl 0820.76008)] it is known that dissipative hyperbolic PDEs with \(d\ge 2\) do not possess the spectral mapping property (contrary to the case \(d=1\)), in general. Hence, in the case \(d\ge 2\) one cannot hope to see Hopf bifurcation like in the case \(d=1\), in general. For the proofs the authors use Fourier series representation of the solutions like in their previous paper [J. Dyn. Differ. Equations 31, No. 1, 129--152 (2019; Zbl 1421.35021)], which concerns the case \(d=1\).
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 58E09
Mathematics Subject Classification ID: 35L71
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35C10
zbMATH DE Number: 7044104
zbMATH Keywords: time delay
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: symmetry breaking
zbMATH Keywords: symmetric group action
zbMATH Keywords: Diophantine equation
zbMATH Keywords: symmetry assumptions
zbMATH Keywords: special frequency vector
MaRDI profile type: Publication
full work available at URL: dcds.2019102
OpenAlex ID: W2909936438
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Wikidata QID: Q128530272
DOI: 10.3934/DCDS.2019102
description: scientific article; zbMATH DE number 7044104