On the Global Solution and Invariance of nonlinear Constrained Modified Swift-Hohenberg Equation on Hilbert Manifold

In this paper, we are interested in finding the existence and uniqueness of the local, local maximal, and global solutions of the equation projected on the Hilbert manifold. Furthermore, we will show that for any given initial data in the Hilbert manifold $\mathcal{M}$, the solution to this equation is also in the Hilbert manifold $\mathcal{M}$. Finally, we will show that the solution to the equation is a gradient flow.