Self-excited vibrations for damped and delayed higher dimensional wave equations
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 \(d=1\).
scientific article; zbMATH DE number 7044104