What question did this study set out to answer?

February 19, 2026Open Access

Blow up dynamics for the 3D energy-critical nonlinear Schrödinger equation

Key Points

To construct and analyze type II blow up solutions for the energy-critical nonlinear Schrödinger equation in three dimensions.
Developed a two-parameter continuum of blow up solutions.
Defined the form of solutions involving energy bubbles.
Studied the time evolution of solutions with parameters lambda and W.
Established that solutions collapse to a single energy bubble in finite time.
Characterized the structure of solutions using specific mathematical forms.
Identified the behavior of the function λ(t) as it approaches critical time T.

Abstract

Abstract We construct a two-parameter continuum of type II blow up solutions for the energy-critical nonlinear Schrödinger equation in dimension d = 3 d = 3. The solutions collapse to a single energy bubble in finite time and have the form aligned u (t, x) = e^i (t) (t) ^1{2}W ( (t) x) + (t, x), ~ ~ t [0, T), ~ x {\, {R\, }}³, aligned u (t, x) = e i α (t) λ (t) 1 2 W (λ (t) x) + η (t, x), t ∈ [ 0, T), x ∈ R 3, where W (x) = (1 + |x|²3) ^-1{2} W (x) = (1 + | x | 2 3) - 1 2 is the ground state solution, (t) = (T -t) ^- 1{2 - } λ (t) = (T - t) - 1 2 - ν for suitable

Demander à l'IA

Bookmark

View Full Paper