$L^\infty$-decay estimate of the solutions to dissipative nonlinear Schrödinger equations, and asymptotic behavior of the solutions

Source abstract

We study the $L^\infty$-decay estimate and the asymptotic behavior of solutions to dissipative nonlinear Schrödinger equations. In some known results, Kita-Nakamura established an $L^{\infty}$-decay result under a weak dissipative condition for nonlinear power $p \gt 2.686$. In this paper, we extend the lower bound of the admissible range to $p \gt 5/2$, still under weak dissipation and with small initial data in $H^2$ satisfying $x^2 u_0 \in L^2$. Our result also contrasts with the work of Jin-Jin-Li, Chinese mathematicians, who treated the same problem under a strong dissipative condition and removed the size restriction on the initial data. Notably, we have solved the problem of extending the nonlinear power $p$ range for the modified scattering and decay estimates without the strong dissipative condition.