Existence and asymptotic behavior for \(L^2\)-norm preserving nonlinear heat equations

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