Existence Of Global Solutions For A Class Of Nonhomogeneous Quasilinear Hyperbolic Equations

PDEs is very important not only in mathematics its own,but also in the other aspects of the natural science.In this paper, we discuss the Cauchy problem for the inhomogeneous quasi-linear hy-perbolic equations. However, the non-homogeneous term is no longer solely dependent on the unknown vectors u, but also on the time variable t.On the basis of two estimations of professor Zhou Yi and normalized coordinates and matching conditions, we can get the decomposition formula of wave under the weak linear degradation.At the same time,we can get the global existence of the solutions.In Chapter l,we introduce the partial differential equation and the quasi-linear hyper-bolic equations is very important in other subjects, and introduces the research is meaningful about quasi-linear hyperbolic equations and expounds the predecessors’s contributions in this field, and then put forward the work of this paper.In Chapter2. we mainly introduce some commonly concepts and definitions in the re-search process.For example, strictly hyperbolic, weak linear degradation, normalized coordi-nates, matching conditions, etc., and give the main problem of this paper.In the Chapter3,we deduced the wave decomposition formula, which not only depends on the time variable t, but depends on the unknown variable u.It plays an important role in the proving of the theorem analyst.In the Chapter4,we mainly refer two estimates and set the stage for the theorem proved.In the Chapter5, we use the prior estimate method to prove the global solution of the in-homogeneous quasi-linear hyperbolic equations is existed under the weak linear degradation, which through the wave decomposition formula and two estimates.