Summary: We consider a semilinear elliptic equation in a cylinder of variable cross-section subject to zero conditions on the lateral boundaries. A second-order differential inequality is obtained for an \(L^{2p}\) cross- sectional measure of the solution, where \(p\) is a positive integer. It is used to obtain an upper bound for the measure in terms of data, supposed specified on the plane ends of the cylinder (finite cylinder). A semi- infinite cylinder is then considered – the principal concern of the paper – and propositions are proved therefor; a global solution, when it exists, must decay at least exponentially in both cross-sectional and energy measures. These results, obtained without assuming that the solution tends to zero at large distances, depend crucially upon a lemma derived from the basic second-order differential inequality.