Convexity and all‐time C ∞ ‐regularity of the interface in flame propagation

We consider the following one‐phase free boundary problem: Find ( u , Ω) such that Ω = { u > 0} and$$\cases{u_t=\Delta u & in $\Omega \cap Q_T$ \cr u=0,|\nabla u|=1, & at $\partial \Omega \cap Q_T$\cr u(x,0)=u_o(x) & on $\Omega_o$.}$$ with Q T = ℝ n × (0, T ). Under the condition that Ω o is convex and log u o 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 < T c , with T c 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.