Periodic Solutions of Non-Dissipatively Perturbed Wave Equations in Several Space Variables.
Abstract
Rabinowitz proved the existence of 2 pi-periodic solutions of a one-dimensional non-linear wave equation for epsilon sufficiently small and f 2 pi-periodic in t, monotone in u, and sufficiently smooth. This answered a long-standing open question ad suggested that monotone methods could be used to overcome solvability problems in bifurcation situations with infinite dimensional kernels. In this paper the methods of Rabinowitz are extended to higher space dimensions, indicating that the special properties of the one-dimensional wave equation are not essential for that result. What remains crucial are hypotheses of rationality in the relations between the time period and the periods of the free vibrations for the wave equation, so that the inverse of the wave operator remains bounded on the complement of the null space. The other crucial factor is the assumption that the non-linearity depends monotonically on u, which enables us to solve for the piece of the solution lying in the (possibly infinite dimensional) null space of the wave operator.
Document Details
- Document Type
- Technical Report
- Publication Date
- Sep 01, 1982
- Accession Number
- ADA124367
Entities
People
- Robert L. Sachs
Organizations
- University of Wisconsin–Madison