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

Abstract

This paper is concerned with the initial–boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation in the half-space R + (I) { u t − u t x x − u x x + f ( u ) x = 0 , t > 0 , x ∈ R + , u ( 0 , x ) = u 0 ( x ) → u + , as x → + ∞ , u ( t , 0 ) = u b . Here u ( t , x ) is an unknown function of t > 0 and x ∈ R + , u + ≠ u b are two given constant states and the nonlinear function f ( u ) ∈ C 2 ( R ) is assumed to be a strictly convex function of u. We first show that the corresponding boundary layer solution ϕ ( 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 ϕ ( x ) are also obtained for both the non-degenerate case f ′ ( u + ) < 0 and the degenerate case f ′ ( u + ) = 0 .