Global Stability And Blow-up Problem Of Solutions For Some Nonlinear Hyperbolic Equations

Hyperbolic equations is an important contents of partial differential equation (PDE) . Studies to hyperbolic equations will promote the further development of PDE theory and other branches of mathematics. The main contents of this thesis consist of two parts. The first one is to study the blow-up of solutions for wave equations with nonlinear damping and source terms. We do this by applying Potential Well theory and Sobolev space theory. The second one is to study global existence and energy decay for wave equations. We do this by combining Lyapunov energy method and Potential Well theory.This thesis consists of three chapters.In Chapter 1, firstly, a survey on the research background and the research advance of the related work are given. Secondly, the main results obtained in this thesis are listed.Chapter 2 is devoted to the study on the properties of solutions for some hyperbolic systems. The properties include local existence, global existence and blow-up of solutions. Hyperbolic systems include Cauchy problem for viscoelastic wave equation, wave equation with a fractional boundary dissipation, and an anisotropic wave equation.In Section 1 of Chapter 2, we consider the following Cauchy problem with viscoelastic term where m≥2,p≥2.g:R+→R+is a C1-function satisfying The initial data u0,u1 and the parameters m,p satisfy the following assumptions with compact support.(G3) 2< p<2n-1/n-2, if n≥3, and p>2, if n=1,2.Assume that the initial energy is negative. Under some suitable assumptions on the kernel function and the parameters in the equation, we establish a finite-time blow-up result and a global existence result, respectively. Section 2 of Chapter 2 is devoted to the study on the following wave equation with fractional derivative term on a part of its boundary WhereΩis a bounded open subset of Rn(n≥1), with a smooth boundary (?)Ω. And (?)Ω=Г0∪Г1,Г0∩Г1=(?),whereГ0 andГ1 are measurable over (?)Ω, endowed with the (n-1)-dimensional Lebesgue measureλn-1(Гi),i=0,1. v is the unit outward normal to (?)Ω. The function f(u)=|u|p-2u is a polynomial source, p>2. The function ia a weakly singular kernel, where 0<α<1,β,b>0. The convolution term in the problem represents a modified fractional derivative of u(in the sense of Caputo).Assume that the initial energy is positive. By using the potential well theory and con-cavity method, we prove that under some suitable assumptions on the parametersα,β,p in the equation, the solution of the problem blows up in finite time.Section 3 of Chapter 2 deals with the following anisotropic nonlinear hyperbolic equation Where pi≥2, i=1,…,n, T> 0,Ωis a bounded open subset of Rn(n≥1), with a smooth boundary (?)Ω,g(u)=u|u|σ-2, (σ>1) is a polynomial source.Under some restriction on the parameters and the initial data, we obtain several results on the local existence of the solution and the blow-up of solutions.Chapter 3 is devoted to studying the second main content:the study on global existence and energy decay of solutions for wave equations with nonlinear damping and source terms.In Section 1 of Chapter 3, we consider the following viscoelastic wave equation Here m≥2, p≥2.Ωis a bounded open subset of Rn(n≥1), with a smooth boundary (?)Ω=Г0∪Г1,Г0∩Г1=(?), where F0 and F1 are measurable over (?)Ω, endowed with the (n-1)-dimensional Lebesgue measureλn-1(Гi),i=0,1.νis the unit outward normal to (?)Ω.上. g is a positive kernel function.By using the potential well theory and Lyapunov energy method, we prove that, under some appropriate assumptions on the kernel function and the parameters, the solution of the problem exists globally and the energy has a general decay.Section 2 of Chapter 3 is concerned with the global existence of solutions for the following quasilinear wave equation: whereΩis a bounded domain of RN. with a smooth boundary (?)Ω.We make the following assumptions:σis a function which satisfies for s>0,The nonlinear damping term has the formThe source term has the polynomial form where the parameters p satisfies: The nonlinear strain termσ(s) satisfies for s≥0, where ai,bi, di,(i=1,2) are nonnegative constants, and b1+b2>0.By using differential inequality and continuation principle, and analyzing the param-eters in the equation, some new sufficient conditions for global existence of the solution were obtained.In Section 3 of Chapter 3, we investigate the following coupled quasilinear wave equation where m,r≥1.Ωis a bounded open domain of RN, with a smooth boundary (?)Ω.We make the following assumptionsσis a C1 function which satisfies for s>0, and for all s≥0, where b1, b2 are nonnegative constants, and b1+b2>0.The source terms f1, f2 have the form and where a, b>0, p≥3. And the initial data u0, u1, v0,v1 satisfy By using differential inequality and continuation principle, and analyzing the relationship between the growth orders of the nonlinear strain term, the damping term and the source term, some new sufficient conditions for global existence of the solution were obtained.