Transition fronts of combustion reaction-diffusion equations in domains with multiple cylindrical branches

This paper is concerned with propagation dynamics for combustion reaction-diffusion equations in domains with multiple cylindrical branches.

We first establish the existence and uniqueness of a time-increasing entire solution behaving like planar traveling fronts in some branches and converging to $0$ in the remaining part of the domain as $t\to-\infty$.

Under the assumption of complete propagation, we then show that this entire solution propagates into the other branches in the form of planar traveling fronts (up to finite shifts) and converges to $1$ elsewhere as $t\to+\infty$.

In particular, it is proved that this entire solution is a transition front connecting $0$ and $1$, whose global mean speed coincides with the planar wave speed.

By assuming complete propagation for front-like solutions originating from single branch, we further prove that every transition front connecting $0$ and $1$ propagates completely.

Moreover, we show that the global mean speed is independent of the choice of transition front.

Namely, all transition fronts connecting $0$ and $1$ share the same global mean this http URL, we provide two sufficient geometric conditions under which the complete propagation assumptions are satisfied.