Damped Nonlinear Schrödinger Equation with Stark Effect

The problem of singularity formation for damped NLS (dNLS) has been an interesting and meanwhile challenging one in both mathematical and physical literature. We study the \(L^2\) -critical damped NLS with a Stark potential. We prove that the threshold for global existence and finite time blow-up of this equation is given by \( \left\| Q \right\| _2\) , where Q is the unique positive radial solution of \(\Delta Q+ |Q|^{4/d}Q=Q\) in \(H^1(\mathbb {R}^d)\) . Moreover, in any small neighborhood of Q, there exists an initial data \(u_0\) above the ground state such that the solution flow admits the log-log blow-up speed. This verifies the structural stability for the “ \(\log \) - \(\log \) law” associated to the NLS mechanism under the perturbation by a damping term and a Stark potential. The proof of our main theorem is based on the Avron-Herbst formula and the analogous result for the unperturbed dNLS. The method of our analysis allows to further prove a general blow-up criterion. Moreover, we give a concentration compactness description for the limiting behavior of blow-up solutions, which might have independent analytical interest.