| In the framework of potential well,by employing the functional analysis,Galerkin method and the cancave method,this thesis considers the finite time blow up phenomena of solutions for the initial boundary value problem to a class of fourth order viscous wave equation at supercritical initial energy level,the well-posedness for some fourth order strain wave equation with logarithmic source terms,strong dissipative term and nonlinear weak damping term,as well as certain stochastic wave equation with nonlinear polynomial source terms,which aims to reveal the relationship between the qualitative properties of the solution and the initial data,and analyze the effect of different media on the natural of solutions.A study on the finite time blowup phenomena for a class of fourth order wave equation with viscous damping term at supercritical energy level by applying the improved concave method.By constructing the monotonicity of the auxiliary function,this chapter proves the invariance of the unstable set with respect to time for supercritical initial energy,which helps us to establish the finite time blow up with a upper bound of blowup time of the solution with supercritical initial energy.A study on the initial boundary value problem for a class of fourth order strain wave equation with logarithmic term,strong damping term and nonlinear weak damping term at three different initial energy levels.Firstly,the existence and uniqueness of the solution for corresponding linear ordinary differential equation are obtained by using the Galerkin method and constructing the approximate solution,which gives the local existence and uniqueness of solution with the aid of Contraction Mapping Principle.Further,the corresponding potential wells are introduced to obtain energy functionals,potential energy functionals,Nehari functionals,the depth of potential well and related properties.Under some appropriate conditions on the initial data,with the boundedness principle,concave method and functional analysis,this chapter proves the global existence of the solution at subcritical initial energy level and critical initial energy level.Moreover,we gain the finite time blow up of the solution at supercritical initial energy level.A study on the initial boundary value problem for a class of stochastic wave equations with nonlinear polynomial sources term.Firstly,the local existence and uniqueness of the solution are established by Galerkin approximation method,which further is proved to be global.Later,by using the idea of energy inequality and concave function method as well as potential well theory,we prove that the corresponding local solution blows up with positive probability in the energy sense. |
