ASYMPTOTIC AND OTHER ESTIMATES FOR A SEMILINEAR ELLIPTIC EQUATION IN A CYLINDER

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 L2p 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.