The Large Time Behavior In Scalar Inhomogeneous Conservation Law With Periodic Initial Data

The large time behavior of the hyperbolic conservation laws is one of the most important problems instudying the solutions of conservation laws. And it has become the focus of mathematicians in recentyears. P.Lax[1]and Dafermos[3]had been studied the large time behavior for the general hyperbolicconservation laws and made a great series of results while H.Fan and J.K.Hale[23]had devoted tostudy the inhomogeneous conservation laws with periodic initial data and proved that the solutionmay approach a travelling waves or a constant. Based on H.Fan and J.K.Hale[23], we generalized theresults of A.N.Lyberopoulos[25]and proved that if the initial data is symmetric with the odd points ofsource term in this paper, then the solutions convergent to a travelling wave. Since the classiccharacteristic theory is no longer applicable to our problem in the weak sense, it needs to extend theconcept of the characteristic, namely generalized characteristics. In this paper, we employ thegeneralized characteristic to investigate the large time behavior of the solutions. The main contentsand the results of this paper are listed as follows:Firstly, we briefly describe the research background and others results, then introduce our maintheorem.Secondly, we introduce the concept of generalized characteristics for using later.Thirdly, we give the weak solution of inhomogeneous conservation laws and get some basicproperties of it under some assumptions about the initial data.Finally, Based on the former preliminary we completed the proof of our main theorem and showan example for it.