Traveling Curved Waves in Two-Dimensional Excitable Media

This paper treats a free boundary problem in two-dimensional excitable media arising from a singular limiting problem of a FitzHugh--Nagumo-type reaction-diffusion system. The existence and uniqueness up to translations of two-dimensional traveling curved waves solutions is shown. To study the stability of the waves, the local and global existence and uniqueness of solutions to the free boundary problem near the waves under certain assumptions is established. The notion of the arrival time is introduced to estimate the propagation speed of solutions to the free boundary problem, which allows us to establish the asymptotic stability of traveling curved waves by using the comparison principle. It is also pointed out that the gradient blowup can take place if the initial data are far from the traveling curved waves, which means the interface may not always be represented by a graph.

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