Existence and multiplicity results for wave equations with time-independent nonlinearity

Abstract

We shall study the existence of time-periodic solutions for a semilinear wave equation with a given time-independent nonlinear perturbation and small forcing. Since the distribution of eigenvalues of the linear part varies with the period, the solvability of the problem depends essentially on the frequency. The main idea of this paper is to consider the situation where the period is not prescribed and hence treated as a parameter. The description of the distribution of eigenvalues as a function of the period enables us to show that under certain conditions the interaction between the nonlinearity and the spectrum of the wave operator induces multiple solutions. Our basic new result states that the autonomous equation admits at least two nontrivial solutions (free vibrations) for a restricted (but infinite) set of periods such that the nonlinearity interacts with one simple eigenvalue. As a corollary we prove that the semilinear wave equation with time-independent nonlinearity and small forcing admits an infinite sequence of pairs of periodic solutions with corresponding period tending to zero. The results are obtained via generalized topological degree theory.

Finite mechanical proxies for a class of reducible continuum systems

“…In this section we apply the ACZ reduction for a class of nonlinear wave equations on the n-dimensional torus. Many results exist in literature for the nonlinear vibrating string ( [9,12,17,18,19,31,52] among others), but only few deal with the higher dimensional case [53,54] using approaches analogous to ACZ.…”

“…The approach and the results proposed here rely on the Amann-Conley-Zehnder reduction (ACZ in what follows), a global Lyapunov-Schmidt technique [9,10,11,12,13,14,15] which transforms infinite dimensional variational principles into equivalent finite dimensional functionals. This method has been employed in conjunction to topological techniques, e.g., Conley index, Morse theory, Lusternik-Schnirelmann category and degree theory, for proving results of existence and multiplicity of solutions for nonlinear differential equations, in particular for semilinear Dirichlet problems, Hamiltonian systems and nonlinear wave equations [9,10,12,13,14,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33].…”