In this paper, we consider the Cauchy problem for the nonlinear Schrödinger equations. In particular, we show the existence of asymptotically free solutions for the dissipative nonlinear Schrödinger equations under mass supercritical setting \(p\ge 1+4/n\) in \(n\ge 1\) space dimensions with data which belong to the weighted Sobolev space \(H^s_2\cap {\mathcal {F}} H^\gamma _2\) for some \(s, \gamma \in (0,1]\cap (0,n/2).\) In previous paper Hoshino (J Differ Equ 266:4997–5011, 2019), the existence of asymptotically free solutions for the dissipative nonlinear Schrödinger equations for some \(1+4/(n+2\gamma )<p<1+4/n\) in \(n\ge 1\) space dimensions with data which belong to the weighted Lebesgue space \({\mathcal {F}}H^\gamma _2\) for some \(0<\gamma \le \min (n/2,1)\) has been studied. The problem for supercritical case remains unresolved. Moreover we show the existence of final state \(\phi _+ \in H^s_2\cap {\mathcal {F}}H^\gamma _2\) and which is not equal to zero; \(\phi _+\not =0\) . This fact gives us new knowledge.
