We study an optimal shape design problem for two-dimensional fractional heat equations driven by the fractional Laplacian \((-\Delta )^s\), \(0<s<1\), in the presence of distributed source terms. The control variable is the spatial domain \(\Omega \subset \mathbb {R}^2\), while the state is governed by a fractional parabolic equation subject to homogeneous exterior Dirichlet conditions. Admissible domains are chosen within the geometric framework introduced by Šverák, which ensures compactness with respect to the complementary Hausdorff topology. A time-dependent shape optimization problem is considered, where the objective functional measures the time-averaged deviation of the state from a prescribed target profile. The associated stationary optimization problem is formulated in terms of the equilibrium state corresponding to the asymptotic source term. Using variational methods, uniform parabolic and elliptic energy estimates, and compactness properties of admissible domains, we establish the existence of optimal domains for both the finite-horizon and stationary problems. For \(\frac{1}{2}<s<1\), we investigate the large-time behavior of optimal designs. We first prove stabilization of the fractional parabolic equation toward its associated stationary state and then establish the convergence of the parabolic cost functional toward the stationary cost functional. Furthermore, we prove a \(\Gamma\)-convergence result with respect to the complementary Hausdorff topology and show that every accumulation point of finite-horizon optimal domains is an optimal domain of the stationary problem. As a consequence, the optimal values converge and a turnpike-type behavior is obtained at the level of optimal shapes. These results extend recent one-dimensional results to a genuinely two-dimensional nonlocal framework and provide a rigorous connection between fractional diffusion, shape optimization, \(\Gamma\)-convergence, and long-time asymptotic analysis.
