The Large Time Behavior In Scalar Balance Law

Partial differential equations is an important method for us to study natural phenomenon anddiscovery its rules. And hyperbolic conservation laws is a significant branch of partial differentialequations. In recent years, lots of mathematicians are getting more and more interest in studying thelarge time behavior of the solutions of hyperbolic conservation laws, and they have achieved abundantresults. For example, M.Shearer and Dafermos[29]had been studied the convex hyperbolicconservation laws. Besides, they pointed out that the solution of the initial value problem collapsed infinite time to a single shock wave joining u_to u+when the initial function is different constantsbeyond a bounded interval. We generalized the results of M.Shearer and Dafermos[29]to theinhomogeneous convex balance laws and proved that the solution collapsed in finite time to a singleshock wave joining G(t+F(u_))toG(t+F(u+))when the initial function is different constantsbeyond a bounded interval. In this paper, we employ the generalized characteristics to investigate thisproblem. The main contents and results of this paper are listed as follows:Firstly, we describe the scientific background of hyperbolic conservation laws, then introduceour main theorem.Secondly, we introduce the concept of generalized characteristics of inhomogeneous balancelaws.Thirdly, we prove that solution of the initial value problem collapsed in finite time to a singleshock wave when the initial function is different constants beyond a bounded interval.Finally, based on the chapter3, we proved our main theorem.