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We study the $\omega$-limit set of solutions of a nonlocal ordinarydifferential equation, where the nonlocal term is such that the space integralof the solution is conserved in time. Using the monotone rearrangement theory,we show that the rearranged equation in one space dimension is the same as theoriginal equation in higher space dimensions. In many cases, this propertyallows us to characterize the $\omega$-limit set for the nonlocal differentialequation. More precisely, we prove that the $\omega$-limit set only containsone element.

On the $\omega$-limit set of a nonlocal differential equation: application of rearrangement theory