Nicholas D. Alikakos

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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 Rn, 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 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 AllenCahn(More)
Several chemical and biochemical processes are typically described by nonlinear coupled partial differential equations “PDE” and hence by distributed parameter models see 1 and the references within . The source of nonlinearities is essentially the kinetics of the reactions involved in the process. For numerical simulation as well as for control design(More)
We rewrite the system ∆u − Wu(u) = 0, for u : R n → R n , in the form div T = 0, where T is an appropriate stress-energy tensor, and derive certain a priori consequences on the solutions. In particular, we point out some differences between two paradigms: the phase-transition system, with target a finite set of points, and the Ginzburg–Landau system, with(More)
We consider the system ∆u−Wu(u) = 0, for u : R → R, W : R → R, where Wu(u) = ( Wu1(u),Wu2(u) ) , in an equivariant class of functions. We prove that there exists u, a two-dimensional solution, which satisfies the conditions u(x1, x2)→ a±, as x1 → ±∞, where a, a− ∈ R are the two global minima of the potential W . We also consider the problem on bounded(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)
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)