Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity (Q525555)

The authors investigate the dynamics of the semilinear wave equation \[ {{\partial^2 u}\over{\partial t^2}} - {{\partial^2 u}\over{\partial x^2}} + \lambda u + h(u) = 0 \] subject to the Dirichlet-periodic boundary conditions \[ u(0, t) = u(\pi, t) = 0, \qquad u(x, t) = u(x, t+2\pi). \] Stating quite weak conditions on the nonlinearity \(h(u)\) the authors prove the existence of \(L^\infty\) solutions tending to \(+\infty\), when the bifurcation parameter \(\lambda\) approaches eigenvalues \(\lambda_0\) of finite multiplicity of the wave operator. The proof applies Lyapunov-Schmidt reduction to the decomposition of the image space into the infinite-dimensional Null space of the wave operator, the finite-dimensional kernel at \(\lambda=\lambda_0\), and of the remaining infinite-dimensional space of eigenfunctions for \(\lambda \not\in \{0, \lambda_0\}\). It would be interesting to know, how the possible non-monotonicity of the nonlinearity \(h(u)\) affects the bifurcation scenario.