Cauchy Problem On A Semi-Bounded Initial Axis For Quasilinear Hyperbolic Systems

In this Ph.D.thesis,we systematically study the Cauchy problem on a semibounded initial axis for quasilinear hyperbolic systems.The global existence,the asymptotic behavior(based on the global existence),the blow-up phenomenon(the lifespan and formation of singularities) of classical solutions to the Cauchy problem are discussed respectively.We also give some examples to show the applications of the theory.The structure of the thesis is as follows:A brief introduction on the research history of classical solutions to the Cauchy problem(both on the whole and a semi-bounded initial axis) for quasilinear hyperbolic systems is given in Chapter 1.The main results of this thesis will be also included in this chapter.For the sake of completeness,in Chapter 2,we give some preliminaries,which contain generalized normalized coordinates,John's formula on the decomposition of waves and three important lemmas to be used in this thesis.In Chapter 3,we consider the global existence of the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.Under the assumptions that the rightmost(resp.leftmost) eigenvalue is weakly linearly degenerate and the inhomogeneous term satisfies the corresponding matching condition,we obtain the global existence and uniqueness of classical solution with small and decaying initial data.Based on Chapter 3,Chapter 4 deals with the asymptotic behavior of classical solution to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.We prove that,when t tends to the infinity,the solution approaches a combination of C^I travelling wave solutions with algebraic rate(1 + t)^-μ,provided the initial data decay with the rate(1 + x)^-(1+μ)(resp.(1 - x)^-(1+μ)) as x tends to +∞(resp.-∞),whereμis a positive constant.Chapter 5 is devoted to the study of the blow-up phenomenon of classical solutions to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems.Under the assumptions that the rightmost(resp.leftmost) eigenvalue is not weakly linearly degenerate and the inhomogeneous term satisfies the corresponding matching condition,for small and decaying initial data satisfying a special condi-tion,we obtain a sharp estimate and the mechanism of formation of singularities of classical solutions.In Chapter 6,we will give some examples to show applications of the theory we study in Chapter 3,Chapter 4 and Chapter 5.