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

Abstract

We investigate the asymptotic behavior of solutions of the initial boundary value problem for the generalized KdV–Burgers equation u t + f ( u ) x = u x x − u x x x on the half-line with the boundary condition 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 ± ) . In the cases of 0 ≤ f ′ ( u − ) < f ′ ( u + ) , we prove the global existence of solutions and asymptotic stability of the weak rarefaction waves when the initial disturbance is small. Also, we can get asymptotic stability of the strong rarefaction waves when f ( u ) satisfies a certain growth condition.