The existence and nonexistence of periodic solutions to certain nonlinear hyperbolic equations (Q2714721)

The author considers some high order hyperbolic problems of the type NEWLINE\[NEWLINEH(u)- F(u)=0NEWLINE\]NEWLINE where \(u=u(t,x)\) with \((t,x)\in \mathbb{R}\times \Omega\), \(\Omega\) an open subset of \(\mathbb{R}^N\), and periodic in \(t\). \(H\) and \(F\) are integro-differential operators on the \(t\) and \(x\) variables respectively.NEWLINENEWLINENEWLINEHere the existence and nonexistence of nontrivial periodic solutions in \(t\) is analyzed. Both existence and nonexistence are analyzed in two stages: first, local equations are considered (\(F\) and \(G\) only depend on derivatives on \(u\)) and second nonlocal equations (the case where integral terms can appear).NEWLINENEWLINENEWLINEThe study of existence of solutions combines separation of variables and a variational technique in the operator \(F\). Either a minimization to obtain at least one solution or Ljusternik-Shnirel'man theory to obtain infinitely many. For this purpose the homogeneity of the coefficients is needed.NEWLINENEWLINENEWLINEThe study of nonexistence is done using certain integral identities. These identities were obtained first by \textit{V. Mustonen} and the author in [Differ. Integral Equ. 11, 133-145 (1998; Zbl 1004.35091)] for radial problems.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00006].