Large-data $L^2$-decay for attractive-dissipative nonlinear Schrödinger equations without the strong dissipative condition

Technical Summary: Large-Data $L^2$-Decay for Attractive-Dissipative Nonlinear Schrödinger Equations

Problem Statement
This paper addresses the large-data $L^2$-decay problem for the nonlinear Schrödinger equation (NLS) with an attractive-dissipative power nonlinearity:
$$i\partial_t u + \frac{1}{2}\Delta u = \lambda |u|^{p-1}u, \quad (t, x) \in (0, \infty) \times \mathbb{R}^d,$$
where the complex coefficient $\lambda$ satisfies $\text{Re } \lambda < 0$ and $\text{Im } \lambda < 0$. The condition $\text{Im } \lambda < 0$ ensures $L^2$-dissipation, while $\text{Re } \lambda < 0$ introduces an attractive (focusing) component.

The primary objective is to establish sharp decay rates for the $L^2$-norm, $\|u(t)\|_2$, for arbitrary initial data $u_0$ in the weighted energy space $\Sigma = H^1(\mathbb{R}^d) \cap \mathcal{F}H^1(\mathbb{R}^d)$. The natural range for this decay problem is $1 < p \leq 1 + 2/d$.

Background and Difficulty
Previous works by Hayashi, Li, and Naumkin [11] established the sharp decay rates (1.2) under the "strong dissipative condition" $(p-1)|\lambda| \leq (p+1)|\text{Im } \lambda|$. When $\text{Re } \lambda \geq 0$ (repulsive-dissipative), the standard energy $E(u) = \frac{1}{2}\|\nabla u\|_2^2 + \frac{\text{Re } \lambda}{p+1}\|u\|_{p+1}^{p+1}$ is sign-definite, providing a uniform bound on the gradient $\|\nabla u(t)\|_2$ and allowing the derivation of decay rates [9].

However, in the attractive-dissipative case ($\text{Re } \lambda < 0$), the standard energy is not sign-definite. Consequently, it fails to directly control the gradient norm $\|\nabla u(t)\|_2$. Recent attempts to overcome this [8, 18] relied on iteration arguments, where preliminary $L^2$-decay estimates are substituted into energy estimates to improve gradient bounds. These methods were limited to restricted ranges of $p$, specifically $p \leq 1 + \frac{4}{3d}$. The range $1 + \frac{4}{3d} < p \leq 1 + \frac{2}{d}$ remained open for arbitrary data.

Methodology
The authors introduce a novel approach that bypasses the iterative arguments used in previous literature. The core of the methodology involves the construction of an augmented energy:
$$E_{\text{aug}}(u) = E(u) + \gamma \|u\|_2^2,$$
where $\gamma > 0$ is a suitable constant depending on the size of the initial data.

1. Augmented Energy Argument: The time derivative of the added term $\gamma \|u\|_2^2$ generates a negative contribution $-2\gamma |\text{Im } \lambda| \|u\|_{p+1}^{p+1}$. This term is sufficiently strong to absorb the non-coercive part of the standard energy estimate arising from $\text{Re } \lambda < 0$.
2. Direct Uniform Bound: By choosing $\gamma$ appropriately (via a scaling argument for large data), the authors prove that $E_{\text{aug}}(u(t))$ is non-increasing. This yields a direct, uniform-in-time bound for $\|\nabla u(t)\|_2$ without requiring intermediate decay estimates or bootstrap iterations.
3. Decay Derivation: With the uniform gradient bound established, the proof proceeds via standard tools:
   • A virial-type estimate relating the weighted norm $\|xu(t)\|_2$ to the time integral of $\|\nabla u(t)\|_2$.
   • A weighted interpolation inequality (Lemma 2.4) linking $\|u\|_q$ to $\|u\|_2$ and $\|xu\|_2$.
   • Combining these with the $L^2$-dissipative identity to derive a differential inequality for $\|u(t)\|_2$.

Key Results
The paper establishes the following main theorem (Theorem 1.1):

For arbitrary initial data $u_0 \in \Sigma$ with $\|u_0\|_\Sigma \neq 0$, the NLS admits a unique global solution $u \in C([0, \infty); \Sigma)$. The solution satisfies the sharp decay rates throughout the entire range $1 < p \leq 1 + 2/d$:

• Critical Case ($p = 1 + 2/d$):
  $$\|u(t)\|_2 \lesssim (\log(1+t))^{-\frac{2}{(d+2)(p-1)}}$$
• Subcritical Case ($1 < p < 1 + 2/d$):
  $$\|u(t)\|_2 \lesssim (1+t)^{-\frac{2}{(d+2)(p-1)}\left(1 - \frac{d(p-1)}{2}\right)}$$

These estimates hold for the attractive-dissipative case ($\text{Re } \lambda < 0, \text{Im } \lambda < 0$) without imposing the strong dissipative condition (1.1).

Significance and Claims
The authors claim that this work removes the remaining restriction $p \leq 1 + \frac{4}{3d}$ that previously limited the understanding of large-data decay in the attractive-dissipative regime.

• Removal of Iteration: The primary technical contribution is the elimination of the iteration argument required in prior works [8, 18]. By using the mass dissipation itself to restore coercive energy control via the augmented energy, the authors obtain a direct uniform gradient bound.
• Generality: The result applies to arbitrary data in $\Sigma$, contrasting with other related works that often require additional smoothness, spatial decay, or structural conditions on the initial data.
• Scope: The method is robust enough to apply to the repulsive-dissipative case ($\text{Re } \lambda \geq 0$) as well, recovering the same decay rates without the strong dissipative condition, though the standard energy already suffices for the gradient bound in that scenario.

In summary, the paper resolves the open problem of $L^2$-decay for large data in the full sharp range $1 < p \leq 1 + 2/d$ for attractive-dissipative NLS by introducing an augmented energy functional that directly controls the gradient norm.