| In this paper, we consider the Cauchy problem for the generalized Korteweg-de Vries- Burgers equation on R x R+ Ut + f(uX p.u + with initial date U1t0 = uo(x) ? u, u < u as 2 ? ?0, where p.> 0, 8 € R, f is a smooth convex function defined on R. Under the assumption that u < u, we study the Cauchy problem mentioned above from two aspects as follows via the P-energy method. 1. The existence of global smooth solution and the asymptotic behavior of the solution as t ? . Roughly speaking, there exists a unique global smooth solution u(z,t) to the above Cauchy problem with some suitable restriction to the initial data. Moreover, the solution u(z, t) satisfies sup u(, t) ?uR(r/t) ? 0 as t ? oo, where uR(/t) is the cen- tered rarefaction wave of the non-viscous Burgers equation Ut + f(u) 0 with Riemann initial data tLt桹 uta</sup> f <sub> < o, a > 0. 2. The convergence of the solution sequence {uI?(z,t)} with respect to parameters p. and 8. i.e, i4a,t) ? ?,t) as p.,& ? 0, where ?z,t) satisfies the following problem ut眆(u),=0, { It=o uo() 4 U, Z 4 This paper is made up of four parts. Part one introduces the background of Korteweg-de Vries-Burgers equaiotn and the relevant research progress. Furthermore, we state our main results following from some retrospection of the results obtained by previous mathematicians. Part two introduces some known preliminary results which will be used frequently in the following text. After smoothing the solution to the rarefaction wave we further 3 MASTER S THESIS give some new decay estimates which plays an important role in the following P-energy estimate. Part three states the stability theory of the rarefaction wave, i.e throrem 3.3, which is based on the local existence and a prior estimate. Part four proves that the Korteweg-de Vries-Burgers equation is a viscous approxima- tion of the nonviscous Burgers equation.
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