Self-excited vibrations for damped and delayed 1-dimensional wave equations
This nice paper concerns 1-dimensional damped and delayed semilinear wave equations of sine-Gordon type with homogeneous Dirichlet boundary conditions. Taking the time delay as bifurcation parameter it is shown that there exists a sequence of Hopf bifurcations from the trivial stationary solution, and a formula for the corresponding bifurcation direction is given. The authors apply the Lyapunov-Schmidt method and the implicit function theorem by working in Sobolev spaces on the two-torus and with corresponding Fourier expansions and by using some special symmetry properties of the nonlinearities. This approach is well-known for ODEs and parabolic PDEs, but for hyperbolic PDEs there appear essential and deep technical difficulties connected with the derivative loss property (small divisors) of hyperbolic PDEs. For hyperbolic systems without delay, but with general nonlinearities, these difficulties have been overcome already by \textit{I. Kmit} and \textit{L. Recke} [J. Differ. Equations 257, No. 1, 264--309 (2014; Zbl 1310.35030), and ibid. 259, No. 11, 6287--6337 (2015; Zbl 1343.35023)].
scientific article; zbMATH DE number 7047662
scientific article