In this paper, we consider a class of quasilinear equations defined on a three-dimensional semi-infinite cylinder, in which the solutions of the equations are assumed to satisfy the nonhomogeneous conditions on both the finite end and the side of the cylinder. An expression of "energy" is defined. By limiting the nonlinear terms and making full use of the differential inequality technique, we obtain a first order differential inequality. By solving this inequality, we prove the alternative results, i.e., the "energy" increases exponentially (polynomial) or decays exponentially (polynomial) with the distance from the finite end. Finally, in the case of decay, we get the upper bound of total energy.