Propagation profile of solutions of multistable reaction-diffusion equations with V-shaped initial values

Curved fronts play an important role in the study of propagation phenomena of bistable reaction-diffusion equations in high-dimensional space. It is well known that in two-dimensional space, the propagation profile of solutions of bistable reaction-diffusion equations with V-shaped initial values can be characterized by a V-shaped traveling front. In this paper, we further investigate multistable reaction-diffusion equations in two-dimensional space. For any given $ \theta\in (0, \frac{\pi}{2}) $, we first give the definition of a V$ _\theta $-shaped propagating terrace, which is a stacked family of V-shaped traveling fronts. Then, we show that the solutions of the Cauchy problem with V-shaped initial values will (quasi-) converge to the V$ _\theta $-shaped propagating terrace as the time sequence goes to infinity.