Global Existence Or Blowup Of Classical Solutions To Nonlinear Multidimensional Hyperbolic Equations

The research on nonlinear hyperbolic equations is not only an important basic problem in partial differential equations, but also admits some strong physical back-grounds, such as in fluid dynamics, quantum mechanics, cybernetics and so on. At present, there are some well known mathematicians in this filed:J.Bourgain (winner of Fields Prize in 1998), Tao Terence (winner of Fields Prize in 2004), S.Alinhac (Pro-fessor of University of Paris XI), S.Klainerman (Professor of Princeton University), I.Rodnianski (Professor of Princeton University) and so on. As for the nonlinear wave equations with small initial values in IRn: where and f vanishes of second order at 0, when u0, u1 ∈ C0∞, though many mathematicians’efforts, we know that equations(0.0.3) have global smooth solutions for n> 3 or blow up for n=1,2,3.It is noted that the functions f and gjk in (0.0.3) are the ones which do not depend on the solution u itself but only its derivatives. When f and gjk depend on u, the situations will become very complicated, that is, the property of solution u will depend on the structure of f and gjk seriously. For example, S. Alinhac in 2003 proved the global existence of the solution to such a nonlinear wave equation where n= 3 and c(u)= 1+u. His work is very technical and contains many new ideas. In 2008, H. Lindblad gave a same result for more general c(u) by another method which comes from some geometric techniques in the general relativity.Besides the important equation (0.0.4), there are many classical nonlinear wave equations containing the solution u with strong physical backgrounds, such as some quasilinear wave equations from fluid dynamics (for example, the pressure-gradient model αt1u-div(euVu)=0) and general relativity, nonlinear wave equation with special structures from material science (liquid crystal equation αt2-c(u)div(c(u)Vu)= 0) and variational wave equation.In this thesis, we will consider the more general nonlinear wave equations (0.0.3), where gjk and f are the smooth functions of both u’ and u. Since we only consider the classical solutions, we always assume that the initial data are smooth and have compact support in this thesis, and then we can get the blowup solution or global existence according to the different types of equations when applying different ways respectively. They are listed as follows:In Chapter 2, we focus on the special kind of nonlinear wave equations:3-D pressure-gradient system. This system arises from the splitting of the three-dimensional compressible Euler system, which can be reduced into such a nonlinear wave equation αt2v-div(evVv)=0, where div( and x= (xi, x2,x3). In this chapter, we will establish a blowup result of classical solutions to the 3-D pressure-gradient systems when the initial data are of small perturbations with respect to the constant states.In chapter 3, we mainly concern with the small data smooth solution problem of such a 3-D nonlinear wave equation:αt2u-(1+u+αtu)△u=0, which is a typical representation of the more general forms = 0, where x0=t, V= (α0,d1,...,α3), gij(u,Vu) is smooth on its arguments and can be certainly written as with cij,dij and eijk being some constants, and dij≠0 for some (i,j), moreover u does not fulfil the null condition. For the 3-D nonlinear wave equations αt2u-(1+u)△u=0 or αt2u-(1+ δtu)△u= 0, H. Lindblad, S. Alinhac, and F. John have proved the global existence or blowup of small data smooth solutions. In this chapter, we will mainly show that the small data smooth solution of αt2u- (1+u+-αtu)△u =0 will blow up in finite time and an explicit expression of lifespan Te is also established when the initial data are radially symmetric.In chapter 4, we pay our attention to the general 2-D quasilinear wave equation =0, where gij(u,Vu) is same as in chapter 3. When the null condition is not fulfilled, we will show that the small data smooth solution blows up in finite time, moreover, an explicit expression of lifespan and blowup mechanism are also established.In chapter 5, we consider the global smooth symmetric solution to 2-D full com-pressible Euler system of Chaplygin gases. For one dimensional or multidimensional compressible Euler system of poly tropic gases, it is well known that the smooth solu-tion will generally develop singularities in finite time. However, for three dimensional Chaplygin gases, due to the crucial role of "null condition" in the potential equation which is derived by the irrotational and isentropic flow, P.Godin in has proved the glob-al existence of a smooth 3-D spherically symmetric flow with variable entropy when the initial data are of small smooth perturbations with compact supports to a con-stant state. In this chapter, we will focus on the global symmetric solution problem of 2-D full compressible Euler system of Chaplygin gases. Motivated by the methods in [4], where the global small data smooth solution is shown when a 2-D quasilinear wave equation satisfies both null conditions (i.e., the first and second null conditions), through finding an appropriate "ghost weight" and carrying out rather involved anal-ysis we can derive some uniform weighted energy estimates on the small symmetric solution to 2-D compressible Euler system of Chaplygin gases and further establish the global existence of the smooth solution by continuous induction method.