On uniqueness and continuation properties after blow‐up time of self‐similar solutions of nonlinear schrödinger equation with critical exponent and critical mass

Source abstract

Abstract We consider the nonlinear Schrödinger equation with critical power where u : (0, T ) × ℝ N → C and ø ϕ H 1 ∪ {ø;|x|ø ϵ L 2 }. With this nonlinear term, the equation (1)‐(1′) has a conformal invariance. Thus one yields explicit “self‐similar” solutions which have the following property: they have minimum L 2 norm among blow‐up solutions. In this paper, we first focus on uniqueness properties of these self‐similar solutions in a certain class. We then look at the possible continuations of these solutions after the blow‐up time.