Existence and asymptotic behavior for L²-norm preserving nonlinear heat equations

We consider a nonlinear parabolic equation with a nonlocal term which preserves the L² 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 H¹ H 1 . Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H¹ H 1 to a stationary state. For a ball, we prove strong convergence to the ground state when the initial condition is positive.