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The Quarterly Journal of Mechanics and Applied Mathematics 1992 45(4):617-627; doi:10.1093/qjmam/45.4.617
© 1992 by Oxford University Press
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ASYMPTOTIC AND OTHER ESTIMATES FOR A SEMILINEAR ELLIPTIC EQUATION IN A CYLINDER

J. N. FLAVIN, R. J. KNOPS and L. E. PAYNE

( Department of Mathematical Physics, University College Galway
Department of Mathematics, Heriot-Watt University Riccarton, Edinburgh EH14 4AS
Department of Mathematics, Cornell University Ithaca, New York 14853, USA )

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.


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