Stable Self-Similar Blow-Up Dynamics for Slightly L 2 Super-Critical Nls Equations

Abstract

We consider the focusing nonlinear Schrodinger equations i∂ t u + Δu + u|u|p−1 = 0 in dimension 1 ≤ N ≤ 5 and for slightly L 2 super-critical nonlinearities \({{p_{c} < p < (1 +\varepsilon)p_{c}}}\) with \({{p_{c} = 1+\frac{4}{N}}}\) and \({{0 < \varepsilon \ll 1}}\) . We prove the existence and stability in the energy space H 1 of a self-similar finite-time blow-up dynamics and provide a qualitative description of the singularity formation near the blow-up time.