Existence and asymptotic behavior for $L^2$-norm preserving nonlinear heat equations

TL;DR

This paper investigates a nonlinear heat equation with a nonlocal term that preserves the $L^2$-norm, analyzing existence, uniqueness, and long-term behavior of solutions on bounded domains and in Euclidean space.

Contribution

It provides new results on local and global well-posedness in $H^1$ and establishes conditions for strong asymptotic convergence to ground states.

Findings

Weak convergence in $H^1$ to stationary states

Strong convergence to ground state for positive initial data in a ball

Analysis of asymptotic behavior in different domains

Abstract

We consider a nonlinear parabolic equation with a nonlocal term, which preserves the $L^2$-norm of the solution. We study the local and global well posedness on a bounded domain, as well as the whole Euclidean space, in $H^1$. Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H^1 to a stationary state. For a ball, we prove strong asymptotic convergence to the ground state when the initial condition is positive.