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+R+equation(I){ut−utxx−uxx+f(u)x=0,t>0,x∈R+,u(0,x)=u0(x)→u+,asx→+∞,u(t,0)=ub. Here u(t,x)u(t,x) is an unknown function of t>0t>0 and x∈R+x∈R+, u+≠ubu+≠ub are two given constant states and the nonlinear function f(u)∈C2(R)f(u)∈C2(R) is assumed to be a strictly convex function of u  . We first show that the corresponding boundary layer solution ϕ(x)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)u(t,x) to the above initial–boundary value problem toward the boundary layer solution ϕ(x)ϕ(x) are also obtained for both the non-degenerate case f′(u+)<0f′(u+)<0 and the degenerate case f′(u+)=0f′(u+)=0.