The Existence Problem Of Multiple Periodic Solutions To Nonlinear Wave Equations

In this thesis,we mainly study the existence problem of multiple periodic solutions to nonlinear wave equations.On the one hand,we consider the periodic solutions of the constant coefficients wave equation with radial symmetry in an n-dimensional ball.For any dimension n,when the period T and the radius R satisfy 8R/T=a/b?where a and b are relative prime?,we prove the existence of at least three radially symmetric periodic solutions for a class of nonlinear problems by using the variational method and saddle point reduction technique.On the other hand,we consider the periodic solutions of the one-dimensional wave equations with variable coefficients.Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in non-isotropic media.For such model,different types of nonlinear terms are discussed respectively.?1?For the nonlinear term with asymptotically linear growth conditions,under some conditions??_??x?>0?that the coefficients satisfy,we study the properties of wave operators with Dirichlet boundary conditions,prove the existence of essential spectrum and characterize the existence interval of essential spectrum,and then obtain the existence of at least three periodic solutions via variational method and saddle point reduction technique.?2?For the nonlinear term with sublinear growth conditions,under some con-ditions??_??x?>0?that the coefficients satisfy,we study the properties of wave operators with some homogeneous boundary conditions,find the existence of essen-tial spectrum,characterize the existence interval of essential spectrum,and then prove the existence of infinitely many periodic solutions by using Z₂-index theory and approximation method.?3?For the nonlinear term with superlinear growth conditions,we study the spectral properties of wave operator for Dirichlet-Neumann and Dirichlet-Robin boundary conditions,and with the aid of minimax principle and approximation method,we prove the existence of infinitely many periodic solutions.In particular,in this case,we do not need to impose any restrict conditions on the coefficients.