Existence and stability of curved fronts for spatially periodic combustion reaction-diffusion equations in ${\mathbb{R}}^N$

This paper is concerned with curved fronts of combustion reaction-diffusion equations in spatially periodic media in ${\mathbb{R}}^N$ $(N\ge 2)$. Under the assumption that there are moving pulsating fronts for any given propagation direction $e\in {\mathbb{S}}^{N-1}$, and by constructing suitable super- and sub-solutions, we prove the existence of a curved front with polytope-like shape in ${\mathbb{R}}^N$. Then we show that the curved front is unique and asymptotically stable.