On the Boussinesq-Burgers equations driven by dynamic boundary conditions

We study the qualitative behavior of the Boussinesq-Burgers equations on a finite interval subject to the Dirichlet type dynamic boundary conditions. Assuming H1×H2 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.