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We consider the following one-phase free boundary problem: Find (u, Ω) such that Ω = {u > 0} and $$\cases{u_t=\Delta u & in $\Omega \cap Q_T$ \cr u=0,|\nabla u|=1, & at $\partial \Omega \cap Q_T$\cr u(x,0)=u_o(x) & on $\Omega_o$.}$$ $ u_t=\La u & \mbox{in $\Omega \cap Q_T$}\\ u=0, \ |\D u|=1, & \mbox{at $\partial \Omega \cap Q_T$}\\ u(x,0)=u_o(x) & \text{on $\Omega_o$}.$--> with QT = ℝⁿ × (0, T). Under the condition that Ωo is convex and log uo is concave, we show that the convexity of Ω(t) and the concavity of log u(·, t) are preserved under the flow for 0 ≤ t ≤ T as long as ∂Ω(t) and u on Ω(t) are smooth. As a consequence, we show the existence of a smooth-up-to-the-interface solution, on 0 < t < Tc, with Tc denoting the extinction time of Ω(t). We also provide a new proof of a short-time existence with C^2,α initial data on the general domain. © 2002 John Wiley & Sons, Inc.