# American Institute of Mathematical Sciences

May  2019, 39(5): 2413-2435. doi: 10.3934/dcds.2019102

## Self-excited vibrations for damped and delayed higher dimensional wave equations

 1 Department of Mathematics and Statistics, University of South Alabama, 411 University Blvd North, Mobile AL 36688, USA 2 Department of Mathematics, Wofford College, 429 North Church Street, Spartanburg, SC 29303, USA

* Corresponding author: Nemanja Kosovalić

Received  October 2017 Revised  October 2018 Published  January 2019

In the article [12] it is shown that time delay induces self-excited vibrations in a one dimensional damped wave equation. Here we generalize this result for higher spatial dimensions. We prove the existence of branches of nontrivial time periodic solutions for spatial dimensions $d\ge 2$. For $d> 2$, the bifurcating periodic solutions have a fixed spatial frequency vector, which is the solution of a certain Diophantine equation. The case $d = 2$ must be treated separately from the others. In particular, it is shown that an arbitrary number of symmetry breaking orbitally distinct time periodic solutions exist, provided $d$ is big enough, with respect to the symmetric group action. The direction of bifurcation is also obtained.

Citation: Nemanja Kosovalić, Brian Pigott. Self-excited vibrations for damped and delayed higher dimensional wave equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2413-2435. doi: 10.3934/dcds.2019102
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