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

TL;DR

This paper establishes explicit exponential convergence rates and stability of solutions to a constrained nonlinear heat equation on a manifold, using spectral analysis and the Łojasiewicz-Simon inequality.

Contribution

It provides new sharp exponential stability results and decay rates for solutions of a nonlinear constrained heat flow, with a novel approach combining spectral analysis and Łojasiewicz-Simon inequality.

Findings

Solutions converge exponentially to the ground state in various norms.

Decay rates depend on the Łojasiewicz-Simon exponent.

The approach includes spectral analysis and higher-order estimates.

Abstract

We consider a nonlinear constrained heat flow evolving on the manifold $\mathcal{M}=\{v\in L^{2}:\|v\|_{L^{2}}=1\}$ 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 \L{}ojasiewicz-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 \L{}ojasiewicz-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.