Asymptotic Behavior Of The Solutions For The Initial-Boundary Problem Of The Generalized BBM-Burgers Equation

In this paper, we consider the initial-boundary problem for the generalized Benjamin-Bona-Mahony-Burgers equation (for brevity, we call it BBM-Burgers equation later),Where f(u) is a smooth 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 [10], the signs of the characteristic speeds f'(u±) divide the asymptotic state into five cases:We also get the global existence with some suitable restriction to the initial data. Moreover, for the case (1) and (2), sup |u(x,t) - Φi(x)| → 0 as t →∞, where Φi(x)(i = 1, 2)x∈R+is the stationary solution of (I), which satisfy:For the case (4) and (5), sup|u(x, t) - (:r, t)| → 0, as t → ^, where ^iR(x, t)(i = 4, 5)x∈R+ =uR(x/t)|R+, and uR(x/t)|R+ is the rarefaction wave of the Riemann problemFor the case (3), sup|u(x, t) - Φ3(x,t)|→ 0, as t →^, where Φ3(x, t) is thex∈R+ superposition of Φ2(x) and ψR4(x,t), and φ3(x,t) =φ2(x} +ψR4(x,t).This paper is made up of two chapters.In chapter one, we introduce the background of generalized BBM-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 three sections, In section 1, we study the cases (1) and (2). In section 2, we study the cases (4) and(5). In section 3, we study the case (3). For the five cases, we all get the global existence and asymptotic behavior as t →∞ of the solution for the problem (I).