[Paper Review] Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation

Pascal Bégout|arXiv (Cornell University)|Jul 9, 2012

Advanced Mathematical Physics ProblemsMathematics20 references22 citations

TL;DR

This paper establishes explicit necessary and sufficient conditions for global existence of solutions to the nonlinear Schrödinger equation with supercritical nonlinearity ($\alpha > 4/N$) in the attractive case ($\lambda > 0$). By introducing two threshold functions $\gamma_*$ and $r_*$ with $\gamma_* < r_*$, it proves that solutions are globally bounded if $\|\varphi\|_{L^2} \leq \gamma_*(\|\nabla\varphi\|_{L^2})$, and blow up in finite time if $\|\nabla\varphi\|_{L^2} > r_*(\|\varphi\|_{L^2})$, recovering Weinstein's sharp condition in the critical case $\alpha = 4/N$. The results improve known blow-up criteria and extend global existence criteria beyond the small data regime.

ABSTRACT

In this paper, we consider the nonlinear Schrödinger equation with the super critical power of nonlinearity in the attractive case. We give a sufficient condition and a necessary condition to obtain global or blowing up solutions. These conditions coincide in the critical case, thereby extending the results of Weinstein. Furthermore, we improve a blow-up condition.

Motivation & Objective

• To extend the sharp global existence condition of Weinstein (for $\alpha = 4/N$) to the supercritical regime ($\alpha > 4/N$), where only small data global existence was previously known.
• To derive explicit, computable conditions in terms of $L^2$ and $H^1$ norms that guarantee global existence or finite-time blow-up.
• To improve existing blow-up criteria by showing that any nontrivial solution with nonpositive energy blows up in finite time, even when energy is zero.
• To demonstrate that the set of initial data leading to global solutions remains unbounded in $H^1$, despite the supercritical nonlinearity.

Proposed method

• Derives a sufficient condition for global existence using a variational approach and a Gagliardo-Nirenberg-type inequality with the best constant $C_*$.
• Introduces two threshold functions $\gamma_*$ and $r_*$ derived from the best constant in the Gagliardo-Nirenberg inequality and the ground state solution of an associated elliptic equation.
• Applies conservation of mass and energy to bound the $H^1$ norm of the solution via a functional inequality involving $\|\nabla u(t)\|_{L^2}$.
• Uses a scaling argument and the construction of a family of initial data $\varphi_{a,b}$ via a projection operator $\mathcal{P}$ to prove blow-up for $b > r_*(a)$.
• Employs a contradiction argument based on the positivity of a functional $f(x) = a - x + b x^p$ to show that $\|\nabla u(t)\|_{L^2}$ remains bounded, implying $T^* = \infty$.
• Relies on the existence and properties of the ground state solution $R$ of the elliptic equation $-\Delta R + \omega R = \lambda |R|^\alpha R$ to construct blow-up initial data.

Experimental results

Research questions

• RQ1 Can a sharp global existence condition be established for the supercritical nonlinear Schrödinger equation ($\alpha > 4/N$), analogous to Weinstein's result for the critical case ($\alpha = 4/N$)?
• RQ2 What explicit, computable conditions on the initial data $\varphi$ in terms of $\|\varphi\|_{L^2}$ and $\|\nabla\varphi\|_{L^2}$ guarantee global existence or finite-time blow-up?
• RQ3 Does the energy-based blow-up criterion extend to the supercritical case when energy is nonpositive, even in the zero-energy case?
• RQ4 How does the geometry of the initial data space ($H^1 \cap L^2(|x|^2 dx)$) affect the global behavior of solutions in the supercritical regime?

Key findings

• Solutions are globally bounded in time if $\|\varphi\|_{L^2} \leq \gamma_*(\|\nabla\varphi\|_{L^2})$, where $\gamma_*$ is explicitly defined via the best constant in the Gagliardo-Nirenberg inequality.
• Solutions blow up in finite time for both $t>0$ and $t<0$ if $\|\nabla\varphi\|_{L^2} > r_*(\|\varphi\|_{L^2})$, with $r_*$ also explicitly defined.
• The set $\mathcal{A} = \{\varphi \in H^1(\mathbb{R}^N) \mid \|\varphi\|_{L^2} \leq \gamma_*(\|\nabla\varphi\|_{L^2})\}$ is unbounded in $H^1$, indicating global solutions exist for large initial data.
• The blow-up condition is improved: any nontrivial solution with nonpositive energy ($E(\varphi) \leq 0$) blows up in finite time, even when $E(\varphi) = 0$, under the supercritical assumption.
• The results recover Weinstein's sharp condition in the critical case $\alpha = 4/N$ as $\alpha \searrow 4/N$, showing $\gamma_*$ and $r_*$ converge to the same threshold.
• For any $(a,b) \in (0,\infty)\times(0,\infty)$ with $a > \rho_*(b)$, there exists an initial data $\varphi_{a,b}$ with $\|\varphi_{a,b}\|_{L^2} = a$, $\|\nabla\varphi_{a,b}\|_{L^2} = b$, such that the solution blows up in finite time.