Asymptotic behavior for a dissipative nonlinear Schrödinger equation with time-dependent damping

We investigate the effect of a time-dependent damping on the asymptotic behavior of solutions for the dissipative nonlinear Schrödinger equation \(\begin{aligned} \displaystyle i \partial _{t}u+\Delta u+i\frac{\gamma }{2(1+t)} u=\lambda |u|^\alpha u, \end{aligned}\) $$\begin{array}{c}\hfill {\displaystyle i{\partial }_{t}u+\mathrm{\Delta }u+i\frac{\gamma }{2(1+t)}u=\lambda {|u|}^{\alpha }u,}\end{array}$$ on \(\displaystyle \mathbb {R}^N\) ${\displaystyle {\mathbb{R}}^N}$ where \(t\ge 0,\) $t\ge 0,$ \(\gamma >0\) $\gamma >0$ and \(\lambda \in \mathbb {C}\) $\lambda \in \mathbb{C}$ satisfying \(\operatorname {Im}\lambda <0.\) $Im\lambda <0.$ Assuming \(\displaystyle \frac{2}{N+\gamma +2}<\alpha <\frac{2}{N+\gamma },\) ${\displaystyle \frac{2}{N+\gamma +2}<\alpha <\frac{2}{N+\gamma },}$ we reveal that the time-dependent damping plays a crucial role in shaping the long-time behavior of the solution, particularly the asymptotic profile, the \(L^\infty \) $L^{\infty }$ and the \(L^2-\) $L^2-$ decay rates.