[Paper Review] Lifespan of solutions for the nonlinear Schrödinger equation without gauge invariance

Masahiro Ikeda|arXiv (Cornell University)|Nov 29, 2012

Advanced Mathematical Physics Problems8 references3 citations

TL;DR

This paper establishes an upper bound for the lifespan of solutions to the defocusing nonlinear Schrödinger equation with non-gauge-invariant power nonlinearity in the subcritical regime $1 < p < 1 + 2/n$. Using a weighted energy method and refined estimates involving radial cutoffs and scaling arguments, the authors derive a sharp upper bound $T_\varepsilon \leq C\varepsilon^{1/\kappa}$ with $\kappa = k/2 - 1/(p-1)$, where $k \in (n, 2/(p-1))$, under suitable decay and sign conditions on the initial data's real or imaginary part.

ABSTRACT

We study the lifespan of solutions for the nonlinear Schrödinger equation id_{t}u+Δu=λ|u|^{p}, (t,x)\in[0,T) imesR^{n}, with the initial condition, where 1

Motivation & Objective

• To close the gap between known lower and upper bounds for the lifespan of solutions to the nonlinear Schrödinger equation with non-gauge-invariant nonlinearity.
• To establish a quantitative upper bound on the maximal existence time $T_\varepsilon$ for small initial data in the subcritical regime $1 < p < 1 + 2/n$.
• To analyze the blow-up mechanism in the absence of gauge invariance, where standard global existence results do not apply.
• To refine existing blow-up results by providing explicit estimates on the lifespan, rather than just proving finite-time blow-up.

Proposed method

• A weighted energy method is employed, introducing a radial cutoff function $\chi_R(x)$ to localize the solution in space.
• The method involves deriving a differential inequality for the weighted $L^2$-norm of the solution, using the integral formulation of the NLS equation.
• A scaling argument is applied to transform the problem into a self-similar variable $\tau = t/T_\varepsilon$, enabling asymptotic analysis.
• The analysis relies on constructing a comparison function $H(\tau, R)$ and comparing it to a model function $G(\tau)$ that captures the growth behavior.
• The key step involves choosing parameters such as $k$ and $\kappa = k/2 - 1/(p-1)$ to ensure $\kappa < 0$, which leads to a finite upper bound on $T_\varepsilon$.
• The proof uses contradiction and monotone convergence to derive bounds on the $L^p$-norm of the solution, ultimately yielding the lifespan estimate.

Experimental results

Research questions

• RQ1 What is the sharp upper bound for the lifespan of solutions to the non-gauge-invariant NLS in the subcritical case $1 < p < 1 + 2/n$?
• RQ2 How does the choice of initial data, particularly its decay and sign properties, affect the blow-up time?
• RQ3 Can a quantitative lifespan estimate be derived when the standard global existence theory fails due to lack of gauge invariance?
• RQ4 What is the role of the parameter $k$ in determining the sharpness of the upper bound?
• RQ5 Is the upper bound $T_\varepsilon \leq C\varepsilon^{1/\kappa}$ optimal, and how does it compare to the known lower bound?

Key findings

• An upper bound for the lifespan is established as $T_\varepsilon \leq C\varepsilon^{1/\kappa}$ for small initial data, where $\kappa = k/2 - 1/(p-1)$ and $k \in (n, 2/(p-1))$.
• The upper bound is valid under the condition that either $f_1 \in L^1$ with $\lambda_2 f_1(x) \geq |x|^{-k}$ for $|x| > 1$, or $f_2 \in L^1$ with $-\lambda_1 f_2(x) \geq |x|^{-k}$, for $|x| > 1$, with $n < k < 2/(p-1)$.
• The lifespan estimate is sharp in the sense that $\kappa < 0$, ensuring the bound tends to zero as $\varepsilon \to 0$, consistent with finite-time blow-up.
• The result closes the gap between the known lower bound $T_\varepsilon \geq C\varepsilon^{1/\omega}$ with $\omega = n/4 - 1/(p-1)$ and the new upper bound.
• The method provides a constructive mechanism for blow-up, unlike previous contradiction-based proofs, and yields explicit dependence on $\varepsilon$, $p$, and $\lambda$.
• The analysis confirms that the $L^2$-norm of the solution blows up in finite time under the given conditions, consistent with the blow-up result in [4].