Convergence rates of solutions toward boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equations in the half-space

This paper is concerned with the initial-boundary value problem of the generalized Benjamin-Bona-Mahony-Burgers equation in the half-space R+ {u(t) - u(txx) - u(xx) + f(u)x = 0, t > 0, x is an element of R+, u(0,x) = u(0) (x) -> u(+), as x -> +infinity, u(t,0) = u(b). Here u (t, x) is an unknown function of t > 0 and X E R+, u + 0 ub are two given constant states and the nonlinear function f(U) is an element of C-2(R) is assumed to be a strictly convex function of u. We first show that the corresponding boundary layer solution phi(x) of the above initial-boundary value problem is global nonlinear stable and then, by employing the space-time weighted energy method which was initiated by Kawashima and Matsumura [S. Kawashima, A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985) 97-127], the convergence rates (both algebraic and exponential) of the global solution u (t, x) to the above initial-boundary value problem toward the boundary layer solution phi(x) are also obtained for both the non-degenerate case f'(u(+)) < 0 and the degenerate case f'(u+) = 0. (C) 2008 Elsevier Inc. All rights reserved.