Phragmén-Lindelöf alternative results for the shallow water equations for transient compressible viscous flow

In this paper, the spatial behavior of the viscous shallow water equations in a two-dimensional channel domain is studied. A differential inequality for suitable energy associated with the solutions of the viscous shallow water equations in a semi-infinite channel is derived, from which we show the Phragmén-Lindelöf alternative results. The main tool used is the energy method. Our results can be viewed as a version of Saint-Venant's principle to the transient compressible viscous flow.