In this paper, we consider an optimal design problem for the 1D nonlocal heat equations involving the fractional Laplacian \((-\frac{d^2}{dx^2})^s (0<s<1)\) . The control here is the shape of the interval on which heat diffuses. We work in the geometric setting introduced by Šverák in [1] where the intervals under consideration are assumed to have a limited number of holes. Based on a \(\Gamma\) -convergence approach, we can prove when \(s>\frac{1}{2}\) , the nonlocal parabolic optimal designs converge, in the complementary Hausdorff topology, to an optimal design for the corresponding stationary nonlocal elliptic equation when time tends to infinity.
