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-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.