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We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in R n , when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally(More)
We present a theory that enables us to construct heteroclinic connections in closed form for 2uxx = Wu(u), where x ∈ R, u(x) ∈ R 2 and W is a smooth potential with multiple global minima. In particular, multiple connections between global minima are constructed for a class of potentials. With these potentials, numerical simulations for the vector Allen-Cahn(More)
Mathematically, the problem considered here is that of heteroclinic connections for a system of two second order differential equations of gradient type, in which a small parameter ǫ conveys a singular perturbation. The motivation comes from a multi-order-parameter phase field model developed by Braun et al [5] and [23] for the description of crystalline(More)
We consider the system where W u (u) = W u1 (u), W u2 (u) , in an equivariant class of functions. We prove that there exists u, a two-dimensional solution, which satisfies the conditions u(x 1 , x 2) → a ± , as x 1 → ±∞, where a + , a − ∈ R 2 are the two global minima of the potential W. We also consider the problem on bounded rectangular domains with(More)
and Applied Analysis 3 2. Preliminaries and Main Results In this section, we first make some preliminary works, then we state explicitly our main results. 2.1. Functional Spaces Let ·, · and | · | denote respectively the inner product and norm ofH L2 G . We define the linear operator A −Δ with domain D A H2 G ⋂H1 0 G . A is positive and selfadjoint. By(More)
We clarify a point in [1], concerning the derivation of the Plateau angle conditions for the vector-valued Allen-Cahn equation. The Plateau angle conditions for the vector-valued Allen-Cahn equation have recently been rigorously derived in the interesting preprint [1]. However, it is not clear to the author how the estimates (30) and (31) in [1] are used(More)