THE PERSISTENCE OF LOGCONCAVITY FOR POSITIVE SOLUTIONS OF THE ONE-DIMENSIONAL HEAT-EQUATION

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, w(·, 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]. © 1990, Australian Mathematical Society. All rights reserved.