The Global Existence Of Classical Solutions To Critical Semilinear Wave Equations With Variable Coefficients

In this Ph.D. thesis, we study the global existence of classical solutions to critical semilinear wave equations with variable coefficients. As is well known, there are two key steps to prove global existence for critical semilinear wave equations:first, prove the non-concentration of energy; second, combine the energy non-concentration with Strichartz estimates. In order to prove energy non-concentration, we construct some geometrical multiplier, just like the Morawetz multiplier used in the constant coef-ficients case. If the coefficients depend only on the spatial variable, we use the distance function to construct the multiplier, and then use the classical comparison theory from Riemannian Geometry to deal with the error terms. When the coeffi-cients depend not only on the space but also on the time, the problem becomes more difficult. Since in this case time and space can’t be split and the variable coefficients induce a Lorentz metric instead of a Riemannian metric. Following the work of Christodoulou and Klainerman, we construct the multiplier by using the null frame associated to an optical function, and then use the comparison theory originated from Riemannian Geometry to estimate the more complicated error terms. The difference is:we study the behavior of the solutions approaching to a fixed point while they explore the asymptotic behavior at infinity.In1968, Sattinger proved the global existence of solutions to some nonlinear wave equations when the initial data lie in the potential well and satisfy a certain condition. Inspired by this, we get the global existence and establish the global exact boundary controllability for focusing cublic semilinear wave equations when the initial data satisfy the same conditions, using the constructive method introduced by Zhou and Lei [66]. We arrange this thesis as follows:First of all, a brief survey of the background and the history on the study of global existence to critical semilinear wave equations is given in Chapter1. Also we introduce some results related to the exact boundary controllability for semilinear wave equations.In Chapter2, we prove the non-concentration of energy to exterior problem of three dimensional semilinear wave equations when the variable coefficients depend on the space only. Then combing the Strichartz estimates obtained by H.Smith and C.D.Sogge [52], we get the global existence.In Chapter3, by using a geometrical multiplier induced from the null frame, we prove the energy non-concentration to Cauchy problem of three dimensional semilinear wave equation with variable coefficients depending on both space and time, and furthermore, prove the global existence.In the last chapter, we establish the global existence and global exact boundary controllability for focusing cubic semilinear wave equations when the initial data stay in the potential well and satisfy some conditions.