Blow Up And Uniqueness Of Solutions For Full Variational Sine-Gordon Equations

In this thesis,we study the blow-up of smooth solutions and the uniqueness of conserved energy weak solutions of a class of full variational Sine-Gordon equations.This equation has important applications in the field of physics research,such as nonlinear optics,liquid crystal theory and so on.The smooth solutions of this equation can blow up in finite time,so the study of its well-posedness has important theoretical significance.For the blow up of smooth solutions,firstly,the problem is transformed into a new Cauchy problem by using Riemann invariants.Then the energy equation and the Riemann invariant in the characteristic region are estimated.And finally we show that the Cauchy problem will blow up in a finite time.For the uniqueness of conservation solutions,firstly,we introduce appropriate variables to make a priori estimate of the total energy of the wave interaction.And then the existence and uniqueness of solutions of the characteristic equation can be obtained for arbitrary initial conditions.Furthermore,we prove that the related functions are Lipschitz continuous with respect to the characteristic variables,which satisfy a semi-linear system with smooth coefficients and the solution of the semi-linear system is unique.Finally the uniqueness of solutions of full Sine-Gordon equations is proved.