Convexity of level curves for an initial value problem

Abstract

Let u be a solution of u t = Δu − u 7 with Dirichlet data equal to 1 on the boundary of a bounded convex domain in R n and initial data also equal to 1, where 0 < γ < 1. It is shown that u has convex level curves for all time t. This problem is used as an example to indicate a general technique for showing that the level curves of solutions of suitable initial value problems are convex.