Abstract
We study the long-term behavior of the solution to a nonlocal evolution equation which describes the limit of Ising spin systems with Glauber dynamics and Kac potentials on a bounded domain. We then propose an unconditionally stable, convergent finite difference scheme to this equation. We prove that the scheme is uniquely solvable, inherits the properties of the original equation, and that the numerical solution will approach the true solution in the $L^\infty$-norm.
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