Global Well-posedness For Two Classes Of Wave Equations

In this paper we study a class of nonlinear Klein-Gordon equations with special nonlinear source terms and a class of nonlinear equations with nonlinear combined power-type nonlinearities of different signs.Firstly,we study the global well-posedness of Cauchy problem for a class of nonlinear Klein-Gordon equations with special nonlinear source terms.It is well known that the nonlinear Klein-Gordon equation are very significant nonlinear evolution equations and play a very important role in the area of mathematical physics.Firstly,by handling the inequality construction of special nonlinear source terms,constructing the energy functionals equation and establishing variational structures,we get the local existence of solutions in the low initial energy,the critical initial energy and the sup-critical initial energy,by using energy estimations and boundness of the norms we obtain the global existence of solutions at the three energy levels.Finally,we redefine the proper auxiliary functions employ the potential well theory,concavity method,embedding theorem and properties of the norms to prove a finite time blow up result of solutions.Potential well theory is mainly base on the variation tendency of potential energy,constructing the variational sets in system,thus analyzing meticulously for this motion tendency of system.Next,we focus on the sharp conditions for the global existence of wave equation with nonlinear combined power-type nonlinearities of different signs at arbitrary high initial energy level,which is used mainly to describe dynamical behavior of solutions for system with two nonlinear combined external forces.For this class of complicated wave equations,by constructing the variational structures,defining the functional spaces and estimation of principle of the maximum,we analyze the properties of solutions and show the sufficient condition of global existence result by potential well theory at arbitrary high initial energy.This paper studies the sharp conditions of global existence and global nonexistence.Indeed,the global existence of solutions mean that the physical model can afford influence of external forces.Meanwhile,the global nonexistence of solutions often represent the collapse of systems.So the research on this topic has extremely important practical significance in scientific researches and industrial constructions.