Global solutions and Asymptotic Behavior to a Norm-preserving Non-local Parabolic Flow

Boris Shakarov

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arXiv · 2411.18532

Global solutions and Asymptotic Behavior to a Norm-preserving Non-local Parabolic Flow

Boris Shakarov

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Abstract

We consider a nonlinear parabolic model that forces solutions to stay on a $L^2$-sphere through a nonlocal term in the equation. We study the local and global well-posedness on a bounded domain and the whole Euclidean space in the energy space. Then, we consider the solutions' asymptotic behavior. We prove strong convergence to a stationary state and asymptotic convergence to the ground state in bounded domains when the initial condition is positive.

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Boris Shakarov. 2024-11-27. Global solutions and Asymptotic Behavior to a Norm-preserving Non-local Parabolic Flow. https://arxiv.org/abs/2411.18532

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