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

TL;DR

This paper investigates a nonlinear parabolic equation that preserves the $L^2$ norm, analyzing its well-posedness and long-term behavior, including convergence to stationary and ground states.

Contribution

It introduces a nonlocal term ensuring solutions stay on an $L^2$-sphere and studies their existence, uniqueness, and asymptotic convergence in bounded and unbounded domains.

Findings

Proved local and global well-posedness in energy space.

Established strong convergence to stationary states.

Demonstrated asymptotic convergence to ground states for positive initial data.

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.