This paper is devoted to study the transition fronts of combustion reaction–diffusion equations in spatially periodic media in \(\mathbb {R}^N\) . With the help of the exponentially asymptotic behaviors of pulsating fronts and its derivatives, we first prove that the propagating speeds of transition fronts satisfy some estimates related to the wave speeds of pulsating fronts. Then, we show that there is a new entire solution for combustion reaction–diffusion equations in \(\mathbb {R}^2\) which behaves as two curved fronts as time goes to \(-\infty \) and as a curved front as time goes to \(+\infty \) .
