Properties And Control Problem Of The Solutions For Some Nonlinear Wave Equations

Wave equations have been one of the important contents of partial differential equa-tion (PDE) and control theory. Studies to wave equations will accelerate the development of PDE and control theory. The main contents of this thesis consist of two parts. The first one is to study the properties of the solutions for the wave equations with nonlinear damp and nonlinear source. We do this by applying PDE theory and Sobolev space the-ory. The second one is to study the controllability and energy decay for wave equations with variable coefficients principal part. We do this by combining Riemannian geometry method and Carleman estimates.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 the solutions for some nonlinear vibration systems. The properties conclude global existence, blow-up, nonexistence of global solution.In Section 1 of Chapter 2, we consider the system of nonlinear wave equations whereΩis a bounded domain with smooth boundary (?)Ωin Rn,n= 1,2,3;m, r≥1; fi(·,·):R2→R2 are given functions to be specified later. Assume that the initial energy is negative. Under some suitable assumptions on the functions f1 and f2, the initial data and the parameters in the equations, the theorems of global existence and nonexistence are proved, respectively. Section 2 of Chapter 2 is devoted to the study on the viscoelastic wave equations whereΩis a bounded domain with smooth boundary (?)Ωin Rn,n= 1,2,3;m,r≥1; fi(·,·):R2→R2,i= 1,2, are given functions to be specified later. Assume that the initial energy is positive. Under some suitable assumptions on the functions f1,f2,ρ1,ρ2, g1,g2, parameters r,m and the initial data u0,u1,v0,v1, the global nonexistence theorem for solutions is proved.Section 3 of Chapter 2 deals with the following damped nonlinear beam equation where a>0,b>0, p>1 and m> 1. Suppose that the initial data are given by Suppose that the right end of the beam is hinged, i.e., and at the left the input u(t)= (u1(t),u2(t)) and the output y(t)=(y1(t),y2(t)) are exerted, satisfying and It is proved that, under some conditions, by constructing auxiliary function, the system has global solution and blow-up solution, respectively.Chapter 3 is devoted to the second main goal:the study to the controllability and energy decay for wave equations with variable coefficients principal part. In Chapter 3,Ωis a bounded domain in Rn with smooth boundaryΓ. It is assumed thatΓconsists of two parts:Γ0 andΓ1,Γ0∪Γ1=Γ, withΓ0 nonempty and relatively open inΓ. Let v denote the outward normal vector field along the boundary.In Section 1 of Chapter 3, we do some preliminaries:introduce Riemannian geometry method.Section 2 of Chapter 3 is concerned with exponential decay of the energy of the following problem: Exponential decay of the energy is proved provided that the function f, k and g satisfy some assumptions. In Section 3 of Chapter 3, we investigate energy decay problem of the following wave equation where (?)(x)∈W1,∞(Ω) is a known function, is the derivative of u alongνA= Aν. f is a continuous nonlinear nonnegative function defined onΓ1, and satisfies some assumptions. Under some different assumptions, polynomial decay and exponential decay are obtained, respectively. In Section 4 of Chapter 3, set be a second-order differential operator, with aij= aji of class C1, satisfying for some constant a0>0. We study exact controllability for the following coupled wave equation with variable coefficients principal part where Q=Ω×(0,T],Σ=Γ×(0,T] andΣi=Γi×(0,T],i= 0,1. w1(x, t) and w2(x, t) are control actions on the boundary, andα,βdenote the spring and damper coupling constants, respectively. Applying Riemannian geometry method and new Carleman esti-mates, the observability inequality and exact controllability for the system are obtained.