SELF-EXCITED VIBRATIONS FOR DAMPED AND DELAYED HIGHER DIMENSIONAL WAVE EQUATIONS

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 >= 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.