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

In this paper, we are interested in proving the existence and uniqueness of the local, local maximal, and global solutions of the equation projected on the Hilbert manifold. Furthermore, we 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 demonstrate that the solution to the equation is a gradient flow.