Asymptotic stability of rarefaction waves for the generalized KdV–Burgers equation on the half-line

We investigate the asymptotic behavior of solutions of the initial boundary value problem for the generalized KdV–Burgers equation ut+f(u)x=uxx−uxxxut+f(u)x=uxx−uxxx on the half-line with the boundary condition u(0,t)=u−u(0,t)=u−. The corresponding Cauchy problems of the behaviors of weak and strong rarefaction waves have respectively been studied by Wang and Zhu [Z.A. Wang, C.J. Zhu, Stability of the rarefaction wave for the generalized KdV–Burgers equation, Acta Math. Sci. 22B (3) (2002) 309–328] and Duan and Zhao [R. Duan, H.J. Zhao, Global stability of strong rarefaction waves for the generalized KdV–Burgers equation, Nonlinear Anal. TMA 66 (2007) 1100–1117]. In the present problem, on the basis of the Dirichlet boundary conditions, the asymptotic states are divided into five cases dependent on the signs of the characteristic speeds f′(u±)f′(u±). In the cases of 0≤f′(u−)