On a sharp lower bound on the blow-up rate for the 𝐿² critical nonlinear Schrödinger equation

Source abstract

We consider the L 2 L^2 critical nonlinear Schrödinger equation i u t = − Δ u − | u | 4 N u iu_t=-\Delta u-|u|^{\frac {4}{N}}u with initial condition in the energy space u ( 0 , x ) = u 0 ∈ H 1 u(0,x)=u_0\in H^1 and study the dynamics of finite time blow-up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in L l o c 2 L^2_{loc} a sharp and stable upper bound on the blow-up rate: | ∇ u ( t ) | L 2 ≤ C ( log ⁡ | log ⁡ ( T − t ) | T − t ) 1 2 |\nabla u(t)|_{L^2}\leq C\left (\frac {\log |\log (T-t)|}{T-t}\right )^{\frac {1}{2}} . In an earlier paper, the authors then addressed the question of a lower bound on the blow-up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is, lim t → T T − t | ∇ u ( t ) | L 2 = + ∞ . \lim _{t\to T}\sqrt {T-t}|\nabla u(t)|_{L^2}=+\infty . In this paper, we prove the sharp lower bound $$| ∇ u ( t ) | L 2 ≥ C ( log ⁡ | log ⁡ ( T − t ) | T − t ) 1 2 |\nabla u(t)|_{L^2}\geq C \left (\frac {\log |\log (T-t)|}{T-t}\right )^{\frac {1}{2}}$$ by exhibiting the dispersive structure in the scaling invariant space L 2 L^2 for this log-log regime. In addition, we will extend to the pure energy space H 1 H^1 a dynamical characterization of the solitons among the zero energy solutions.