Generation, motion and thickness of transition layers for a nonlocal Allen–Cahn equation

We investigate the behavior, as ε→0ε→0, of the nonlocal Allen–Cahn equation ut=Δu+1ε2f(u,ε∫Ωu), where f(u,0)f(u,0) is of the bistable type. Given a rather general initial datum u0u0 that is independent of εε, we perform a rigorous analysis of both the generation and the motion of the interface, and obtain a new estimate for its thickness. More precisely, we show that the solution develops a steep transition layer within the time scale of order ε2|lnε|ε2|lnε|, and that the layer obeys the law of motion that coincides with the limit problem within an error margin of order εε.