Refinement of the $L^{2}$-decay estimate of solutions to nonlinear Schrödinger equations with attractive-dissipative nonlinearity
Naoyasu Kita, Hayato Miyazaki, Takuya Sato
TL;DR
This work studies $L^{2}$-decay for nonlinear Schrödinger equations with attractive-dissipative nonlinearity under the dissipative condition $\Im \lambda<0$, without size restrictions on the initial data. The authors strengthen the time-decay analysis by refining an energy-type estimate and leveraging a scale-invariant inequality, yielding global well-posedness in $\Sigma$ for $1<p\le 1+\tfrac{4}{3d}$ and precise decay rates: a logarithmic decay at the critical exponent $p=1+\tfrac{4}{3d}$ and a polynomial decay for $1<p<1+\tfrac{4}{3d}$, matching the HLN16 rate. The approach hinges on obtaining a uniform-in-time bound for $\|x u(t)\|_2$ via virial-type arguments and iteratively improving decay and gradient bounds, thereby extending previous results that required smaller $p$-ranges. The method also applies to the repulsive-dissipative regime, enhancing understanding of long-time behavior in dissipative NLS models and connecting to the best-known decay rates in the literature.
Abstract
This paper is concerned with the $L^{2}$-decay estimate of solutions to nonlinear dissipative Schrödinger equations with power-type nonlinearity of the order $p$. It is known that the sign of the real part of the dissipation coefficient affects the long-time behavior of solutions, when neither size restriction on the initial data nor strong dissipative condition is imposed. In that case, if the sign is negative, then Gerelmaa, the first and third author [7] obtained the $L^{2}$-decay estimate under the restriction $p \le 1+1/d$. In this paper, we relax the restriction to $p \le 1+4/(3d)$ by refining an energy-type estimate. Furthermore, when $p < 1+ 4/(3d)$, using an iteration argument, the best available decay rate is established, as given by Hayashi, Li and Naumkin [11].