Global Well-posedness For Strongly Damped Wave Equations And Coupled Nonlinear Klein-Gordon Equations

This paper settles the initial and boundary of a class of nonlinear wave equations with strongly damping terms and the Cauchy problem of a coupled system of nonlinear KleinGordon equations with nonlinear damping terms.For the initial boundary value problem of a class of nonlinear wave equations with strongly damping terms,we firstly introduce the structure of potential well and related preliminary lemmas.Moreover,we discuss well-posedness of solutions with the low initial energy and critical initial energy respectively.For the low initial energy,we obtain the invariable stable set and unstable set by contradiction arguments firstly,then we obtain the property of global existence and blow up of solutions by using Galerkin method and concavity method respectively.After that we show the asymptotic behavior of the global solution combining the energy functionals by multiplier method.Finally,at arbitrary high initial energy level,we introduce a new stable set and a new unstable set.Under the initial data satisfing some conditions we show that the weak solution exists globally.Meanwhile we prove that the blow up of solutions in finite time by concavity method.For the Cauchy problem of a coupled system of nonlinear Klein-Gordon equations,we define the energy equality and related functionals in the structure of potential well.Using some assumptions of nonlinear source term,we obtain some related lemmas.Moreover,we study the property of global existence and blow up in finite time with the low initial energy and critical initial energy respectively.For the low initial energy in the aid of Galerkin method we have global existence theorem.In order to prove the blow up results,we introduce a proper auxiliary functional,by estimating this auxiliary functional,we prove the function in the growth situation of contradiction,then we prove the property of blow up.For the critical case,we first prove the related character of functionals of J(u,v)and I(u,v),then we obtain the global existence theorem and blow up.Finally,for the arbitrary high initial energy,we confer core method in high initial energy of the above equation,the blow up property in finite time is proved.