Local quasi-concavity of the solutions of the heat equation with a nonnegative potential

In this paper we consider the Cauchy problem for the heat equation with a nonnegative potential decaying quadratically at the space infinity and investigate local concavity properties of the solution. In particular, we give a sufficient condition for the solution to be quasi-concave in a ball for any sufficiently large t, and discuss the optimality of the sufficient condition, identifying a threshold for the occurrence of local quasi-concavity. © 2011 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.