论文标题
解决方案的长期渐近渐近学和用于超临界非线性schrödinger方程的修改的伪符号保护法
Long-time asymptotics of solutions and the modified pseudo-conformal conservation law for super-critical nonlinear Schrödinger equation
论文作者
论文摘要
In this paper, we discuss a class of nonlinear Schrödinger equations with the power-type nonlinearity: $(\mathrm{i} \frac{\partial}{\partial t} + Δ) ψ= λ|ψ|^{2η}ψ$ in $\mathbf R^N \times \mathbf R^+$.基于Gagliardo-Nirenberg的插值不平等，我们证明了解决方案对$（H^Q）$ super-CricationSchrödinger方程的局部存在和长期行为（持续，有限的爆炸或全球存在，连续依赖，连续依赖）。相应的缩放不变空间是均匀的sobolev $ \ dot h^{q_ {crit}} $，带有$ q_ {crit}> q $。基于二次术语的估计值，该术语包含killip，Murphy和Visan \ cite在论文中使用的相衍生物的估计[Siam J. Math。肛门。 50（3）（2018），2681--2739] {kmv018}我们将在解决问题的解决方案上使用更强的束缚来研究稳定性。此外，根据Killip和Visan \ Cite在论文中提出的有关病毒类型的论点[Amer。 J. Math。 132（2）（2010），361--424] {kv010}，提出了修改的伪符合条件保护定律。还提出了摩拉维斯解决问题解决方案的估计。
In this paper, we discuss a class of nonlinear Schrödinger equations with the power-type nonlinearity: $(\mathrm{i} \frac{\partial}{\partial t} + Δ) ψ= λ|ψ|^{2η}ψ$ in $\mathbf R^N \times \mathbf R^+$. Based on the Gagliardo-Nirenberg interpolation inequality, we prove the local existence and long-time behavior (continuation, finite-time blow-up or global existence, continuous dependence) of the solutions to the $(H^q)$ super-critical Schrödinger equation. The corresponding scaling invariant space is homogeneous Sobolev $\dot H^{q_{crit}}$ with $q_{crit} > q$. Based on the estimates of the quadratic terms containing the phase derivatives used in the 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 the solutions to our problem. Moreover, from the arguments on virial-types presented 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. The Morawetz estimate for the solutions to the problem are also presented.
