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The persistence of logconcavity for positive solutions of the one dimensional heat equation

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  09 April 2009

G. Keady
G. Keady
Affiliation:
Mathematics Department, University of Western AustraliaNedlands, W. A. 6009, Australia
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Abstract

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Consider positive solutions of the one dimensional heat equation. The space variable x lies in (–a, a): the time variable t in (0,∞). When the solution u satisfies (i) u (±a, t) = 0, and (ii) u(·, 0) is logconcave, we give a new proof based on the Maximum Principle, that, for any fixed t > 0, u(·, t) remains logconcave. The same proof techniques are used to establish several new results related to this, including results concerning joint concavity in (x, t) similar to those considered in Kennington [15].

MSC classification

Secondary: 35K05: Heat equation
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 2 , April 1990 , pp. 246 - 263
Copyright
Copyright © Australian Mathematical Society 1990

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