Optimal Shape Design of 1D Fractional Order Heat Equation in Large Time

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In this paper, we consider an optimal design problem for the 1D nonlocal heat equations involving the fractional Laplacian . 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 -convergence approach, we can prove when \\frac{1}{2}$$\\end{document}]]>, 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.