### Abstract

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. © 1992 Oxford University Press.

Original language | English |
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Pages (from-to) | 617-627 |

Number of pages | 11 |

Journal | Quarterly Journal of Mechanics and Applied Mathematics |

Volume | 45 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 1992 |

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## Cite this

*Quarterly Journal of Mechanics and Applied Mathematics*,

*45*(4), 617-627. https://doi.org/10.1093/qjmam/45.4.617