In this work, we aim to study the well-posedness of a deterministic problem consisting of the non-linear heat equation of gradient type. The evolution equation emerges by projecting the Laplace operator with Dirichlet boundary conditions and polynomial nonlinearity of degree , onto the tangent space of the sphere in a Hilbert space . We are going to deal with the question of existence and uniqueness based on the Faedo-Galerkin compactness method.