Time Periodic Solutions Of Nonlinear Wave Equation

The wave equation is an important partial differential equation in mathematical physics which generally describes all kinds of waves in the nature, such as sound waves, light waves and water waves. It arises in many different fields, such as acoustics, electromagnetics, and fluid dynamics. Variations of the wave equation are also found in quantum mechanics and general relativity. As the American mathematician Whitham said in [53], almost any field of science or engineering involves of questions of wave motion. Therefore, in order to solve these problems it is natural to study the wave equation which decribes the wave motion. Historically, the problem of a vibrating string such as that of a musical instrument was studied in 18 century by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange, etc. With further research, scholars realized that it is very difficult and hopeless to try to describe fully the behavior which might arise from a nonlinear wave equation. However, it is well known that periodic wave motion is a special but very important physical phenomena, which corresponds to the periodic solutions of wave equation in mathematics. Thus it makes sense to first consider the periodic solutions, in the hope that through a more concrete understanding of them one may gain insight into the behavior of more general solutions. After the renowned French mathematician Poincare [42] first realized the importance of periodic solutions in ordinary differential equaions, there has been an active search for periodic solutions of nonlinear wave equations, employing a variety of methods and motivated, at least in part, by the important role that periodic solutions play in understanding the behavior of ordinary differential equations.The problem of finding periodic solutions of nonlinear wave equations has attracted great attention of many mathematicians since 1960s, such as Rabinowitz, Nirenberg, Brezis, Barbu, Wayne, Craig, Kuksin, Poschel, Bourgain, etc. Until now, plenty of results on the periodic solutions of nonlinear wave equations have been obtained by using of all kinds of methods and techniques, see the survey of Brezis [16], see also the recent references [2, 6, 7, 8, 9, 10, 11, 30, 31, 32, 33, 47, 48]. However, most of them dealt with the wave equations with constant coefficients which well describe the wave motion phenomenon in isotropic media. In the real life, many wave motion phenomenon occur in nonisotropic media, for example, the vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For the decription of thus wave motion, the wave equation with x-dependent coefficients is a much better model. More precisely, as stated in [3, 4, 5, 24, 32, 33, 34, 35, 36, 37], the vertical displacement y(z,t) at depth z and time t of a plane seismic waves is described by the equationunder some boundary conditions in z and initial conditions in t. Hereρis the rock density andμis the elasticity coefficient. By the change of variable given bywe obtainwhere u = (ρμ)^1/2 denotes the acoustic impendance function.Equation (2) is linear and it is relatively simple. Due to the importance in mathematics and physics, people pay more attention to the nonlinear wave equationwhich has become an attractive subject in the fields of mathematical physics and applied mathematics. At the end of the last century, Barbu and Pavel studied the periodic solutions of wave equations with x-dependent coefficients for the first time, see [3, 4, 5]. In [3], they investigated the periodic solutions of (3) under the boundary conditionsfor the case that g = 0 and u(x) is a piece-wise constant function. In the case of smooth coefficients, the same problem as in [3] was considered in [4] along with an inverse problem associated with (3) by them for g = 0. Subsequently, for the case g(x,t,y) = g(y), Barbu and Pavel also studied the periodic solutions of (3) in [5] under the Dirichlet boundary conditionsthey proved the existence and regularity of its periodic solutions provided that the nonlinear term has sublinear growth and satifies the global Lip-schitz condition. Recently, Rudakov [46] also proved the existence of periodic solutions for the case the nonlinear term having power-law growth, i.e., the case that g(x,t,y) = |y|^p-2y with p > 2. Noting that all their results are dealing with the Dirichlet boundary conditions. The natural but very important problems are:Does there exist periodic solutions for equation (3) under the general Sturm-Liouville boundary conditionswhere a_i² + b_i²â‰ 0 for i = 1,2? What properties do they have?Does there exist periodic solutions for equation (3) under the periodic or anti-periodic boundary conditionsAnd what properties do they have?Furthermore, what can we say about the periodic solutions of higher-dimensional wave equation with x-dependent coefficients?All these problems are significant for further understanding the wave motion in nature, so they have attracted extensive attention in the fields of mathematical physics and applied mathematics. Due to the complex nature of the spectrum of wave operator with x-dependent coefficients under general Sturm-Liouville boundary conditions, periodic and anti-periodic boundary conditions, not very much results seem to be known to these problems. In this thesis, we will carry out deep research into these problems. The main results we obtained are as follows:1. Consider the one-dimensional wave equation with x-dependent coefficients under the general Sturm-Liouville boundary conditions. We establish the existence and regularity of periodic solutions of above equation for Dirichlet-Neumann boundary problem, Neuamann boundary problem and general boundary problem, respectively, provided the nonlinear term has sublinear growth and satisfies the global Lipschitz condition. The result has been published in J. Differential Equations 229 (2006) 466-493.2. Under the periodic and anti-periodic boundary conditions, we prove the existence and regularity of periodic solutions of the wave equationfor the nonlinearity having sublinear growth and satisfying the global Lipschitz condition. The result has been published in Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 349-371.3. Consider the one-dimensional wave equationwith p > 2. We prove the existence of periodic solutions of above equation for Dirichlet-Neumann boundary problem, Neumann boundary problem and general boundary peoblem, respectively.4. Under the periodic and anti-periodic boundary conditions, we prove the existence of periodic solutions of the wave equationwhere p > 2.5. Finally, for the two-dimensional wave equation with constant coefficients and two-dimensional wave equation with z-dependent coefficientswe also establish the existence and regularity of periodic solutions under some kinds of boundary conditions.