The Initial-Boundary Value Problem And Cauthy Problem For A Class Of Nonlinear Wave Equation

In this paper, we consider the initial boundary value problem for a class of multidimensional nonlinear wave disperive equation u_tt-â–³u-â–³u_tt=f(u),x∈Ω,t>0, u|_t=0=u₀(x),u_t|_t=0=u₁(x),x∈Ωu|_(?)Ω=0,x∈(?)Ω,t≥0.whereΩ(?)Rⁿ is a propriate bounded domain and f∈C.We also study Cauchy problem of the following equation in Rⁿ u_tt-â–³u-â–³u_tt=f(u),x∈Rⁿ,t>0, u|_t=0=u₀(x),x∈Rⁿ.This thesis mainly consists of three parts:First, this paper introduces the potential well theory. By using the potential well method, a family of potential wells are given.Then, the global existence of solutions, invariant sets, vacuum isolating of solutions and the critical initial condition of the initial boundary value problem are obtained.Last, Cauchy.problem is studied by means of the family of potential wells.