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We study the qualitative behavior of the Boussinesq–Burgers equations on a finite interval subject to the Dirichlet type dynamic boundary conditions. Assuming H 1 × H 2 initial data which are compatible with boundary conditions and utilizing energy methods, we show that under appropriate conditions on the dynamic boundary data, there exist unique global-in-time solutions to the initial-boundary value problem, and the solutions converge to the boundary data as time goes to infinity, regardless of the magnitude of the initial data. [ABSTRACT FROM AUTHOR]