Qualitative Studies Of Solutions For Some Nonlinear Hyperbolic Equations

Hyperbolic equations is an important research contents of the research field of nonlinear partial differential equations.The hyperbolic equations with viscoelastic term and damping term have wide applications in applied disciplines such as materials science,physics,engineering and so on.Then,the studies on hyperbolic equations will advance the further development of nonlinear theory and other disciplines.In this paper,we mainly study the properties of local existence,global existence,decay and finite time blow up of solutions for several kinds of nonlinear hyperbolic equations(systems).In chapter 1,we present the history and current situation for the decay property and the blow-up property of solutions for hyperbolic equations,and the main results obtained in this dissertation.In chapter 2,we introduce the function space,property theorem and some inequalities used in this paper.In chapter 3,we study the initial boundary value problem of a viscoelastic wave equation under the influence of viscoelastic term,weak damping term and source term.By introducing the suitable Lyapunov functional which is equivalent with the modified energy functional,using the perturbed energy method and inequality techniques,we prove that,under some assumptions on the relaxation function and the parameters of the problem,the solution energy has a uniform decay rate with relaxation function.In chapter 4,we discuss the initial boundary value problem for a viscoelastic wave equation with linear weak damping term,strong damping term and source term.Firstly,by using the contraction mapping principle,we analysis the existence and uniqueness of the local solution.Secondly,under the assumption that the initial energy is negative or positive and bounded,we obtain that the solution energy is blow up in finite time under some conditions.Finally,by using potential well theory,we obtain that the solution exists globally in time,and by using the perturbed energy method and inequality techniques,we obtain that the solution energy has a general decay.In chapter 5,we are concerned with the initial boundary value problem for a coupled system of viscoelastic wave equation with the nonlinear weak damping term and coupled source term.Firstly,under some restriction on the relaxation function and the parameters of the system,we prove that the solution is global existence by using the potential well theory.Secondly,we show that the solution energy has a general decay by introducing a suitable Lyapunov functional and using the perturbed energy method and inequality techniques.At last,by establishing a appropriate auxiliary function and using the reduction to absurdity we show the finite time blow up of some solutions whose initial energy is arbitrarily positive.In chapter 6,we analysis the initial boundary value problem of a coupled system of viscoelastic wave equation with dispersion term,nonlinear weak damping term and coupled source term.Under suitable restriction on the relaxation functions and the parameters of the system,we show that the solutions under negative initial energy blow up in finite time.In chapter 7,we give the summary of this paper and the research contents and prospects for future work.