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The mechanisms responsible for pattern formation have attracted a great deal of attention since Alan Turing elucidated his fascinating idea on diffusion-induced instability of steady states. Subsequent studies on the models demonstrated an entirely different class of solutions; namely localized structures composing of steadily moving fronts and pulses. In such energy-driven motion, the combination of short and long-range interaction plays an important ingredient for the generation of complex patterns. This competition on traveling wave dynamics, commonly observed in many physical and chemical phenomena, will be highlighted. [ABSTRACT FROM AUTHOR]

In this paper we consider a bistable reaction-diffusion equation in unbounded domains and we investigate geometric conditions under which propagation, possibly partial, takes place in some direction or, on the contrary, there is a blocking phenomenon. We start by proving the well-posedness of the problem. Then we prove that when the domain has a decreasing cross section with respect to the direction of propagation there is complete propagation. Further, we prove that the wave can be blocked as it comes through an abrupt opening. Finally we discuss various general geometrical properties that ensure either partial or complete invasion by 1. In particular, we show that in a domain that is 'star-shaped' with respect to an axis, there is complete invasion by 1. [ABSTRACT FROM AUTHOR]