Stable periods of a semilinear wave equation and bifurcation of periodic solutions
A semilinear wave equation \[ u_{tt}-u_{xx}=f(\lambda,x,u),\quad t\in {\mathbb{R}},\quad x\in (0,\pi),\quad \lambda \in {\mathbb{R}} \] with Dirichlet or Neumann boundary conditions is studied. It is supposed that f is a smooth function, \(f(\lambda,x,0)=0\), \(f_ u(\lambda,x,0)=a(\lambda)\) (with a(\(\lambda)\) independent of x) for all \(\lambda\in {\mathbb{R}}\), \(x\in <0,\pi >\). The linearized problem has nontrivial periodic solutions (in t) for any \(\lambda\in {\mathbb{R}}\). The question is whether some of these solutions persist in a perturbed form for the nonlinear problem. A set \(\Lambda\) dense in \({\mathbb{R}}\) is introduced such that for any \(a(\lambda_ 0)=a_ 0\in \Lambda\) there are finitely many periods \(P_ r(a_ 0)\) such that nontrivial \(P_ r(a_ 0)\)-periodic solutions bifurcate at \(\lambda_ 0\) from the trivial solution.
scientific article; zbMATH DE number 4023812