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In this paper, we discuss a class of nonlinear Schr\"odinger equations withthe power-type nonlinearity: $(\mathrm{i} \frac{\partial}{\partial t} + \Delta) \psi = \lambda |\psi|^{2\eta}\psi$ in $\mathbf R^N \times \mathbf R^+$. Basedon the Gagliardo-Nirenberg interpolation inequality, we prove the localexistence and long-time behavior (continuation, finite-time blow-up or globalexistence, continuous dependence) of the solutions to the $(H^q)$super-critical Schr\"odinger equation. The corresponding scaling invariantspace is homogeneous Sobolev $\dot H^{q_{crit}}$ with $q_{crit} > q$. Based onthe estimates of the quadratic terms containing the phase derivatives used inthe paper by Killip, Murphy and Visan \cite[SIAM J. Math. Anal. 50(3) (2018),2681--2739]{KMV018} we shall study the stability with a stronger bound on thesolutions to our problem. Moreover, from the arguments on virial-typespresented in the paper by Killip and Visan \cite[Amer. J. Math. 132(2) (2010),361--424]{KV010}, a modified pseudo-conformal conservation law is proposed. TheMorawetz estimate for the solutions to the problem are also presented.

Long-time asymptotics of solutions and the modified pseudo-conformal conservation law for super-critical nonlinear Schr\"odinger equation