Phragmen-Lindelöf decay theorems for classes of nonlinear Dirichlet problems in a circular cylinder

Classes of nonlinear elliptic equations in a long circular cylinder of radius one are considered. The equations are of the form ▽²u = S(u, u′)u″ + T(u)u′², where u = u(x₁, x₂, x₃), and u′, u″ represent general partial derivatives of the indicated order. Homogeneous Dirichlet data are prescribed on the long sides of the cylinder, and throughout the cylinder u is a priori assumed to be sufficiently small while u′ (and, for some classes, also u″) is assumed to be bounded in absolute value by one. With the above assumptions, it is proved that every solution u decays exponentially with distance from the nearer end with a decay constant k which depends on the smoothness properties of S and T but is independent of the length of the cylinder.