Asymptotic behavior of solutions to the generalized BBM-Burgers equation

We investigate the asymptotic behavior of solutions of the initial-boundary value problem for the generalized BBM-Burgers equation u t + f(u) x = u xx + u xxt on the half line with the conditions u(0, t) = u -, u(∞, t) = u + and u - < u +, where the corresponding Cauchy problem admits the rarefaction wave as an asymptotic states. In the present problem, because of the Dirichlet boundary, the asymptotic states are divided into five cases depending on the signs of the characteristic speeds f_(u ±) of boundary state u - = u(0) and the far fields states u + = u(∞). In all cases both global existence of the solution and asymptotic behavior are shown under the smallness conditions. © 2005 Springer-Verlag Berlin Heidelberg.