Rate of convergence of a nonlinear heat equation with a constraint of codimension one
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arXiv
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| Natura: | Preprint |
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2026
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| author | Bawalia, Ashish Mohan, Manil T. |
| author_facet | Bawalia, Ashish Mohan, Manil T. |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10735 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rate of convergence of a nonlinear heat equation with a constraint of codimension one Bawalia, Ashish Mohan, Manil T. Analysis of PDEs 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 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. |
| title | Rate of convergence of a nonlinear heat equation with a constraint of codimension one |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.10735 |