Dynamical properties of constrained heat flow on Hilbert manifolds

We study the long-time dynamics of solutions to a nonlinear gradient flow associated with Problem (2.7), where trajectories are constrained to evolve on a manifold. Using energy methods and spectral properties of the Dirichlet Laplacian, we first establish global existence and precompactness of trajectories in the natural energy space. By proving a Lojasiewicz-Simon inequality for the corresponding energy functional, we deduce convergence of all global solutions to stationary equilibria. Moreover, we provide sharp convergence rates: exponential in the case of nondegenerate equilibria, and polynomial otherwise. Finally, we demonstrate the existence of a compact global attractor in \(\mathcal{V}\cap\mathcal{M}\) that captures the asymptotic behavior of all bounded trajectories. These results place the problem within the general theory of dissipative gradient systems and give a precise description of its asymptotic dynamics. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/13/abstr.html