Rate of convergence of a nonlinear heat equation with a constraint of codimension one

Abstract

We consider a nonlinear constrained heat flow evolving on the manifold $\mathcal{M}=\{v\in L^2:\Vert v{\Vert }_{L^2}=1\}$ over bounded smooth domains. It is known that the solution corresponding to any nonnegative initial datum remains on $\mathcal{M}$ and converges to the unique positive ground state of the associated stationary problem. In this work, we first establish certain time-regularity estimates and then use these to derive explicit exponential rates of convergence for the energy, the solution in the $L^2,H^1$ and $H^2-$norms, and the associated nonlinear eigenvalue, thereby proving a sharp exponential stability of the ground state. Moreover, using the Łojasiewicz-Simon inequality, we obtain decay rates for locally stabilized solutions toward a stationary state in the $L^2$ and $H^1-$norms, where the rate depends on the corresponding Łojasiewicz-Simon exponent. Our results are new, and the approach relies on spectral analysis of the linearized operator, uniform higher-order estimates, and the compactness of solution trajectories.

Appendix A needs to be revisited for clarity and completeness