Electron. J. Differential Equations, Vol. 2026 (2026), No. 13, pp. 1-13.

Dynamical properties of constrained heat flow on Hilbert manifolds

Zdzislaw Brzezniak, Javed Hussain

Abstract:
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.

Submitted November 11, 2025. Published February 11, 2026.
Math Subject Classifications: 335R01, 35K61, 47J35, 58J35.
Key Words: Constrained evolution equations; Lojasiewicz-Simon inequality; global attractor; convergence to equilibrium; decay rate
DOI: 10.58997/ejde.2026.13

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Zdzislaw Brzezniak
University of York
Department of Mathematics
United Kingdom
email: zdzislaw.brzezniak@york.ac.uk
Javed Hussain
Sukkur IBA University
Department of Mathematics
Pakistan
email: javed.brohi@iba-suk.edu.pk

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