Regularity and analyticity in a two-dimensional combustion model. (Q1405957)

The following combustion problem is considered, where \(\Theta, S, \xi\) are related to temperature, mass fraction of the combusting substance, and to the flame front respectively: \[ \begin{aligned} & \Theta_t{(t,\eta,y)}=\Delta\Theta{(t,\eta,y)},\quad t>0,\;y\in \mathbb{R},\;\eta <\xi(t,y)\\ & \Theta(t,\eta,y)=1,\quad t>0,\;y\in \mathbb{R},\;\eta\geq \xi(t,y)\\ & S_t(t,\eta,y)=\Delta S(t,\eta,y)-\lambda\Delta\Theta(t,\eta,y),\quad t>0,\;y\in\mathbb{R},\;\eta\neq\xi(t,y).\end{aligned} \] In a previous paper \textit{C. M. Brauner}, \textit{A. Lunardi} [Arch. Ration. Mech. Anal. 154, No.~2, 157--182 (2000; Zbl 0984.76030)] have shown existence and uniqueness of classical solutions close to a travelling wave solution. In this paper the author obtains finer regularity properties and in particular the analyticity of the free boundary with respect to time under suitable assumptions on the initial data. The technique used is to reformulate the problem as a fully nonlinear evolution problem in a fixed domain and then to set up a procedure enabling to estimate the derivatives of any order.