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The Study Of The Properties For The Solutions Of Several Classes Of Nonlinear Evolution Equations

Posted on:2007-07-04Degree:DoctorType:Dissertation
Country:ChinaCandidate:J H HaoFull Text:PDF
GTID:1100360185950898Subject:Basic mathematics
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Nonlinear partial differential equations (NPDE) is one of the main research fields in nonlinear science. An intensive study of NPDE will accelerate the development of nonlinear analysis. This dissertation focus on the study of the global solutions, blowup solutions and the exact controllability for the solutions of some nonlinear evolution equations.The thesis is composed of five chapters.In chapter 1, the investigation history and the study status of the problems which are discussed in this thesis are listed, and the main results obtained in this thesis are summarized.In chapter 2, we first investigate the following problem of a nonlinear string with nonlinear damping, the boundary input and output,in which p ≥ 2, a > 0, b > 0 are constants, σ(s) is a smooth function, and σ(0) = 0, σ'(s) > 0 for s ∈ R. By constructing suitable auxiliary functions, and using differential inequalities, we prove that under some conditions on the function σ(s), the nonlinear damping term, boundary input u(t) and output y(t), the system has a global solution and a blow up solution, respectively. At the same time, when the system has a blow up solution, we can get the estimate for the blow-up time.Then we study the problem of a nonlinear beam with the nonlinear damping and the two ends of the beam are hinged,in which p ≥ 2, b > 0 are constants, σ(s) is a smooth function, andBy constructing suitable auxiliary functions, we prove that (1) when p > 2 under some conditions on nonlinear damping term, initial energy and the function σ(s), the solution y(x, t) of the system blows up in finite time, and the estimate for the blow up time can be obtained. (2) When p = 2, the solution y(x, t) of the system exists globally in time.In chapter 3, we deal with the boundary exact controllability for the dynamic governed by the beam equation with variable coefficient in the principle part,First we give some inequalities. Next, we prove that the method HUM (Hilbert Uniqueness Method) can be applied to obtain the exact controllability of the dynamic for every T > 0, and the control function v(x, t) has the form v(t) = u_xx(0,t), in which u(x,t) is the solution of the dual system.In chapter 4, we consider an initial-boundary value problem to a vibrating riser with dissipative term,where a, b, α,β are nonnegative constants, f(s) is a continuous function. By constructing suitable auxiliary functions, and using differential inequalities, we prove that under some conditions the solution with negative initial energy blows up in finite time. And we show that the solution with nonnegative initial energy is global in t.In chapter 5, we gvie the following three sections,(1) In section 1, we sonsider the solutions of the nonlinear reaction diffusion equationsviThe study of the properties for the solutionsof several classes of nonlinear evolution equationswith the mixed boundary conditions. By constructing auxiliary functions and using maximum principles, we obtain the sufficient conditions for the blow-up solutions. It is shown that under some assumptions on the functions a(s), b(t), f(x, t, s), the boundary data and the initial data, the solutions exist globally and blow up in a certain finite time, respectively. Moreover, when the solutions blow up we can give the blow-up set, the estimate for the blow-up time and the blow-up rate.(2) Section 2 deals with the solutions of the nonlinear heat equationsUt = Au + m(x, t, u, q)with the mixed boundary conditions, where q = |Vix|2. By constructing auxiliary functions and applying maximum principles to them, thus the sufficient conditions for the global solutions and the blow-up solutions are obtained respectively. At the same time, the bounds of the blow-up time, the blow-up set and the blow-up rate are given.(3) In section 3, we investigate a class of heat conduction equations in a long cylindrical region when the lateral surface and the far end are subjected to a zero temperature. The cylinder is assumed to be at zero temperature initially and subjected to a nonzero temperature distribution at the near end. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the global solutions are obtained. Moreover, the pointwise bounds and spatial decay estimates of the solutions are given.
Keywords/Search Tags:String equation, Beam equation, Heat conduction equation, Global solution, Blow-up solution
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