In this paper, we study the well-posedness of the AdS -sliced domain walls solution for fake N=1 supergravity in higher dimensions. We start with Lagrangian for fake N=1 supergravity which is coupling between gravity and complex scalar in d+1 dimensions with scalar potential turned on. Then, as in supergravity theory, we demand that the scalar fields span the Kahler manifold. The equations of motion for fields can be reduced into first order equations by defining the superpotential and the resulting equations are called the projection equation and the fake BPS equation. Finally, we prove the local existence and uniqueness of the projection and fake BPS equations.
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