Study On The Well-posedness For Some Wave Systems On Cone Manifold

The research topics of my doctoral dissertation is part of the research project "Qualitative studies on nonlinear partial differential equations on singular manifolds(11871017)",which was supported by the National Natural Science Foundation of China.In the framework of potential well theory,we study the well-posedness of global solution for nonlinear wave system with different structural terms at three different initial energy levels,and analyze the relationship between their initial values and the global existence or nonexistence of the solution.Moreover,the finite time blow up of the solution is proved at the arbitrarily positive initial energy level.Based on the framework of potential well,the doctoral dissertation studies the wellposedness of the global solution of four classes of wave equations with different structural terms on the cone manifold for almost all the levels of the initial energy,the characteristics of these four classes of equations,which are classical one,the wave equation with weak damping term,the model with strong damping term and the equation with the potential term respectively.We construct two classes of invariant manifolds which can pass the invariance properties of the initial value to the solution by using the variational structure at the low initial energy level(the initial energy is less than the potential well depth),and then disclose the dependency relationship between the dynamic behavior of the solution and the initial value.As long as the initial value is in the stable manifold,such initial values with these invariance properties lead to the global existence of the solution.Otherwise,the initial values in the unstable manifold lead to the finite time blow up of the solution.For the well-posedness of the global solution at the critical initial energy level,the global solution at the low initial energy level is extended to the case of critical initial energy level by using the conclusion of the global existence of solution at the subcritical energy level and scaling transform methods,and then the finite time blow up of the solution is obtained by using the invariance properties of the unstable manifold.For the arbitrarily positive initial energy level,that is,the initial energy is out of the control of the potential well depth,the invariant manifold parallel to the low energy situation no longer exists,so we need to restrict the positive initial energy so as to obtain the invariant manifold which is affected by the energy structures and methods of the the proof.On the one hand,the positive definite initial energy satisfies different constraints,which requires different methods to prove the finite time blow up of the solution.For example,the classical wave equation and the wave equation with potential term on the cone manifold indicate that as long as the Nehari functional with the initial time is definite negative,the inner product of the initial displacement and the initial velocity can control the arbitrarily positive initial energy,then the solution blows up in finite time.Also we find that when the initial energy is less than some norm of the initial displacement,the finite time blow up of the solution to the problems can be obtained by using the improved concave function method.On the other hand,nonlinear equations with different structural terms produce different energy structures,which combining with the auxiliary functionals,the conditions for initial value satisfied and the controlling of initial energy,jointly decides the existence of invariant manifolds.For the considered four classes of wave equations with different structural terms on the cone manifold,we prove the existence of their invariant manifolds at the arbitrarily positive initial energy level,and the solution generated from the initial value in these invariant manifolds blows up in finite time.To compare the effects of the smooth manifold(Euclidean space)and the cone manifold on the dynamic behavior of the global solution,we also study the one-dimensional and the high-dimensional nonlinear wave equations on smooth manifold.In Chapter 6,we study the existence of the solution for a class of generalized Boussinesq equation with strong damping.We first give the existence of local solution.Then Galerkin method and the concave function method are used to respectively prove the global existence and finite time blow up of the solution at the low initial energy level.Finally,we obtain the global existence of solution at the critical initial energy level.In Chapter 7,we study the existence of global solution for nonlinear KleinGordon equation with combined source terms.Firstly,the global existence and finite time blow up of the solution are proved by using Galerkin and concave function method respectively.Then the global existence and finite time blow up of the solution at the low initial energy level are extended to the case of critical initial energy level.Finally,the finite time blow up of the solution is obtained at the arbitrarily positive initial energy level.