Large-data $L^2$-decay for attractive-dissipative nonlinear Schr\"odinger equations without the strong dissipative condition

TL;DR

This paper establishes a new large-data $L^2$-decay estimate for nonlinear dissipative Schr"odinger equations, removing previous restrictions and providing uniform bounds without iterative methods.

Contribution

It introduces an augmented energy method that yields uniform-in-time $H^1$ bounds for arbitrary data, extending decay results to a broader range of nonlinearities.

Findings

Proves $L^2$-decay for large data without strong dissipative conditions.

Introduces an augmented energy approach for uniform bounds.

Extends decay results to the full sharp decay range $1<p extless 1+2/d$.

Abstract

We prove a large-data $L^2$-decay estimate for nonlinear dissipative Schr\"odinger equations with attractive-dissipative power nonlinearity. The main difficulty is the lack of sign definiteness of the standard energy when $\Re\lambda<0$, which prevents the usual energy argument from directly yielding a uniform gradient bound. We introduce an augmented energy, obtained by adding a suitable multiple of the decreasing $L^2$-norm to the standard energy. This produces an additional dissipative term and gives a direct uniform-in-time $H^1$ bound without the iteration argument used in previous works. Consequently, for arbitrary initial data in the weighted energy space $\Sigma = H^1 \cap \mathcal{F}H^1$, we obtain the decay rate previously known under the strong dissipative condition throughout the sharp decay range $1<p\le 1+2/d$. This removes the remaining restriction $p\le 1+4/(3d)$ in the attractive-dissipative case.