Long Time Solutions Of Nonlinear Wave Equations And Related Problems

In this Ph.D. thesis, we will study some problems in the perturbative theory of nonlinear wave equations. In the first part of this thesis (Chapter 2 and Chapter 3), under the con-ditions of initial data is small and the nonlinear term depends explicitly on the unknown function, we will study lifespan of classical solutions to Cauchy problem of quasilinear wave equations with multiple propagation speeds and initial-boundary value problem of quasilinear wave equations outside of star-shaped obstacles in four space dimensions. The case of four space dimensions just corresponds to the critical case of Strauss conjecture(see the introduction in Section 1.1). Du and Zhou have treated the case of three space dimen-sions in [11] and [10]. In the second part of this thesis(Chapter 4), we will study the local exact boundary controllability of scalar field equations in there space dimensions under the framework of low regularity solutions.The first result of this thesis is that for the Cauchy problem of quasilinear wave e-quations with multiple propagation speeds in four space dimensions, under the conditions of initial data is small and the nonlinear term depends explicitly on the unknown func-tion, we prove the lifespan of the classical solutions Tε≥exp(c/ε2). Our proof is based on Klainerman’s commutative vector fields method(for its history and applications, see [29]). Using the Klainerman-Sideris estimates(a weighted L2 estimate of the second order derivative of the unknown function) which was established in [28] and some basic analysis and Sobolev inequalities, we give a weighted L2 estimate of the first order derivative of the unknown function. We also establish a Lt∞Lx2 estimate of the solution to the linear wave equations. By these key estimates, under the framework of commutative vector field method, we can get the lifespan estimate.The second result of this thesis is that for the initial-boundary value problem of quasilinear wave equations outside of star-shaped obstacles in four space dimensions, under the conditions of initial data is small, the nonlinear term depends explicitly on the unknown function and homogeneous Dirichlet boundary condition, we prove the lifespan of the classical solutions Tε≥exp(c/ε2). Here we will use the modified commutative vector field method which is adapted to exterior problems and first used in [21]. The key point of this method is using the spatial decay efficiently, and the lifespan estimate is implicit in the corresponding estimates of the solution to the linear wave equations. We first get a L∞tLx2 estimate and a weighted Lt2Lx2 estimate for the solution to the Cauchy problem of linear wave equations. Then using the cut-off technique, the corresponding estimate for the exterior problem can be established. As for the estimate for the derivative of the solution, we can use the KSS estimate which was established by Metcalfe and Sogge in [42]. Under the framework in [21], by these key estimates and a Sobolev inequality with spatial decay factor, we can get the lifespan estimate.The third result of this thesis is that under the framework of low regularity solutions, we prove that in the sense of some critical Sobolev norm is sufficiently small, the scalar field equations in three space dimensions is local exact boundary controllable. To prove our result, we will use the constructive method(see [54]), and use the Strichartz estimates which is useful for low regularity problems.The arrangement of the thesis is as follows:In Chapter 1, a brief introduction is given for the history on the study of small initial data problem of nonlinear wave equations, and its relationship with the Strauss conjecture.In Chapter 2, for the Cauchy problem of quasilinear wave equations with multiple propagation speeds in four space dimensions, under the conditions of initial data is small and the nonlinear term depends explicitly on the unknown function, we will prove the desired lifespan estimate.In Chapter 3, for the initial-boundary value problem of quasilinear wave equations outside of star-shaped obstacles in four space dimensions, under the conditions of ini-tial data is small, the nonlinear term depends explicitly on the unknown function and homogeneous Dirichlet boundary condition, we will prove the desired lifespan estimate.Finally, In Chapter 4, under the framework low regularity solution, we will prove the local exact boundary controllability of scalar field equations in three space dimensions.