Asymptotic Stability Of Rarefaction Wave For Generalized Burgers Equation

In this paper, we consider the Cauchy problem for the generalized Burgers equation on R x R+with initial datawhere # > 0, 0 < a < 1/4q(q is determined by (2.2)). / 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:1. The existence of global smooth solution. Roughly speaking, there exists a unique global smooth solution u(x,t) to the above Cauchy problem with some restriction to the initial data. In this paper we only consider the existence of global smooth solution under the initial condition U.₀L鞍. We prove it by the method of successive approximation and maximum priciple.2. The asymptotic stability of the global smooth solution, i.e., the solutionu(x, t) satisfies sup where uR(x/t) is the centeredxeR rarefaction wave of the non- viscous Burgers equation ut + f(u)x = 0 with RiemannThis paper is made up of four parts.Part one introduces the background of the generalized Burgers equation and the relevant research progress. Furthermore, we state our main results following from some retropection 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 give some decay estimates which plays an important role in the following estimate.inPart three states the global existence of solution by maximum principle which based on the local existence of solution given by the method of the successive approximation.Part four proves the asymptotic stability of the rarefaction wave, i.e., Theorem 4.2...