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In this paper, we study entire solutions of the Allen-Chan equation in one-dimensional Euclidean space. This equation is a scalar reaction-diffusion equation with a bistable nonlinearity. It is well-known that this equation admits three different types of traveling fronts connecting two of its three constant states. Under certain conditions on the wave speeds, the existence of entire solutions with merging these three traveling fronts is shown by constructing a suitable pair of super-sub-solutions.

The commonly used spatial entropy hr(U) of the multi-dimensional shift space U is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space Zd, d ≥ 2. This work studies spatial entropy hΩ(U) of shift space U on general expanding system Ω = {Ω(n)}∞ n=1 where Ω(n) is increasing finite sublattices and expands to Zd. Ω is called genuinely d-dimensional if Ω(n) contains no lower-dimensional part whose size is comparable to that of its d-dimensional part. We show that hr(U) is the supremum of hΩ(U) for all genuinely two-dimensional Ω. Furthermore, when Ω is genuinely d-dimensional and satisfies certain conditions, then hΩ(U) = hr(U). On the contrary, when Ω(n) contains a lower-dimensional part, then hr(U) < hΩ(U) for some U. Therefore, hr(U) is appropriate to be the d-dimensional spatial entropy.

This investigation completely classifies the spatial chaos problem in plane edge coloring (Wang tiles) with two symbols. For a set of Wang tiles B, spatial chaos occurs when the spatial entropy h(B) is positive. B is called a minimal cycle generator if P(B) 6= ∅ and P(B′) = ∅ whenever B′ $ B, where P(B) is the set of all periodic patterns on Z2 generated by B. Given a set of Wang tiles B, write B = C1 ∪ C2 ∪ · · · ∪ Ck ∪ N, where Cj, 1 ≤ j ≤ k, are minimal cycle generators and B contains no minimal cycle generator except those contained in C1 ∪ C2 ∪ · · · ∪ Ck. Then, the positivity of spatial entropy h(B) is completely determined by C1 ∪ C2 ∪ · · · ∪ Ck. Furthermore, there are 39 equivalent classes of marginal positive-entropy (MPE) sets of Wang tiles and 18 equivalent classes of saturated zero-entropy (SZE) sets of Wang tiles. For a set of Wang tiles B, h(B) is positive if and only if B contains an MPE set, and h(B) is zero if and only if B is a subset of an SZE set.