Abstract: We considered a class of quasilinear equations defined on a three\|dimensional semi infinite cylinder, in which the solutions of the equations were assumed to satisfy the nonhomogeneous condition at the finite end of the cylinder and the zero boundary condition at the side of the cylinder. By limiting the nonlinear terms and  using  the differential inequality technique, we gave the alternative theorem of the solutions of the equations on three different cylinders, and gave  the upper bound of total energy in the case of decay.

Key words: quasilinear equation, Phragmén-Lindelof type alternative, differential inequality