Existence of Nonplanar Solutions of a Simple Model of Premixed Bunsen Flames

This work deals with the existence of solutions of a reaction-diffusion equation in the plane $\mathbb{R}^2$. The problem, whose unknowns are the real c and the function u, is the following: $$\left\{ \begin{array}{rl} \Delta u-c\displaystyle{{\partial u}\over{\partial y}}+f(u)=0\ & \hbox{ in }\rr^2,\\ \begin{array}{ll}\forall \vec k \in {\cal C} (-\vec{e}_2,\alpha), & u(\lambda \vec k) \displaystyle{\mathop{\longrightarrow}_{\lambda \to +\infty}} 0,\\ \forall \vec k \in {\cal C} (\vec{e}_2,\pi-\alpha), & u(\lambda \vec k) \displaystyle{\mathop{\longrightarrow}_{\lambda \to +\infty}} 1, \end{array} & \end{array}\right.\leqno(P)$$ where $0 < \alpha\le \pi/2$ is given, $\vec{e}_2=(-1,0),$ and, for any angle $\phi$ and any unit vector $\vec e$, ${\cal C}(\vec{e},\phi)$ denotes the open half-cone with angle $\phi$ around the vector $\vec{e}$. The given function f is of the "ignition temperature" type. In this paper, we show the existence of a solution (c,u) of (P) and we give an explicit formula that relates the speed c and the angle $\alpha$.