Convergence Rate And Stability To The Planar Boundary Layer Solutions For BBM-burgers Equations In Two-dimensional Half Space

This paper is concerned with the initial-boundary value problem for the BBM-Burgers equations in the two-dimensional half space. In this paper, u(t, x1, x2) is an unknown function of t ? R+and x =(x1, x2) ?R2+, R2+=R+× R. u+> ubare two given constant states and the nonlinear function f1(u)is assumed to be a non-convex smooth function which has more than one inflection points.It is shown in reference [1, 2] or in reference [3], the above initial-boundary value problem in the half space R+admits a unique global solution u(t, x) which converges to the boundary layer solution ?(x) uniformly in x ? R+as t ? +?, provided that the strict convexity of f(u) or not strict convex of f(u). Moreover the corresponding stability and algebraic convergence rate is obtained.Here we consider the initial-boundary value problem for the BBM-Burgers equations in the high dimensional half space such as in the two-dimensional half space on a weaker condition, that is, we do not ask for the strict convexity of f1(u). By using the principle of image compression, continuity skills and the same space-time weighted energy estimates,for such a case similar stability and the algebraic convergence rate of u(t, x1, x2) toward?(x1) are also is obtained.