Asymptotic Behavior Of The Solutions To The Initial-Boundary Problem Of The Generalized KdV-Burgers Equation

In this paper, we consider the initial-boundary problem for the generalized Kortweg-de Vries-Burgers equation (for brevity, we call it KdV-Burgers equation later),where f(u) is a smooth strictly convex function defined on R and u±are two given constants. Under the assumption of u_- < u_+, we study the global existence and asymptotic behavior of the solution as t→∞for (I), i.e., for the initial-boundary problem (I), as in [14], the signs of the characteristic speeds f'(u_±) divide the asymptotic state into five cases:For cases (4) and (5), we get the global existence , with some suitable restriction to the initial data or f(u) satisfying certain growth condition for large u. Moreover, , as t→∞, where r(x, t) is the rarefaction wave solution of the Riemann problem whereIn this paper, we chiefly discuss the cases (4) and (5), also discuss the case (1) tentatively, and it is made up of three chapters.In chapter one, we introduce the background of generalized KdV-Burgers equation and the relevant research progress. Furthermore, we state our main results following from some retrospection of the results obtained by previous mathematicians.The chapter two is divided into two sections. In section 1, we study the stability of the weak rarefaction waves under small initial data. In section 2, we study the stability of the strong rarefaction waves under large initial data. For the two cases, we all get the global existence and asymptotic behavior of the solution for the problem (I) as t→∞.In chapter three, for case (1), we guess the global existence of the stationary solution of the problem (I), and it decays exponentially. Then, we can get the stability of the stationary solution. At the end of this paper, we give a remark about the cases (1), (2) and (3).