scholarly journals On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kelin Li ◽  
Huafei Di
Keyword(s):  
Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Long Time ◽  
Derivative Term

<p style='text-indent:20px;'>Considered herein is the well-posedness and stability for the Cauchy problem of the fourth-order Schrödinger equation with nonlinear derivative term <inline-formula><tex-math id="M1">\begin{document}$ iu_{t}+\Delta^2 u-u\Delta|u|^2+\lambda|u|^pu = 0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ t\in\mathbb{R} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ x\in \mathbb{R}^n $\end{document}</tex-math></inline-formula>. First of all, for initial data <inline-formula><tex-math id="M4">\begin{document}$ \varphi(x)\in H^2(\mathbb{R}^{n}) $\end{document}</tex-math></inline-formula>, we establish the local well-poseness and finite time blow-up criterion of the solutions, and give a rough estimate of blow-up time and blow-up rate. Secondly, under a smallness assumption on the initial value <inline-formula><tex-math id="M5">\begin{document}$ \varphi(x) $\end{document}</tex-math></inline-formula>, we demonstrate the global well-posedness of the solutions by applying two different methods, and at the same time give the scattering behavior of the solutions. Finally, based on founded a priori estimates, we investigate the stability of solutions by the short-time and long-time perturbation theories, respectively.</p>

scholarly journals GLOBAL WELL-POSEDNESS FOR THE GENERALISED FOURTH-ORDER SCHRÖDINGER EQUATION

2012 ◽  
Vol 85 (3) ◽  
pp. 371-379 ◽  
Author(s):  
YUZHAO WANG
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Vries Equation ◽  
Well Posedness ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Image Position ◽  
Appl Math

AbstractWe study the Cauchy problem for the generalised fourth-order Schrödinger equation for data u0 in critical Sobolev spaces $\dot {H}^{1/2-3/2k}$. With small initial data we obtain global well-posedness results. Our proof relies heavily on the method developed by Kenig et al. [‘Well-posedness and scattering results for the generalised Korteweg–de Vries equation via the contraction principle’, Commun. Pure Appl. Math.46 (1993), 527–620].

scholarly journals The Cauchy Problem for a Weakly Dissipative 2-Component Camassa-Holm System

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Sen Ming ◽  
Han Yang ◽  
Yonghong Wu
Keyword(s):  
Wave Breaking ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Solutions ◽  
Well Posedness ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Blow Up Rate ◽  
Global Existence Result

The weakly dissipative 2-component Camassa-Holm system is considered. A local well-posedness for the system in Besov spaces is established by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation. The wave-breaking mechanisms and the exact blow-up rate of strong solutions to the system are presented. Moreover, a global existence result for strong solutions is derived.

scholarly journals On Jensen-type Inequalities for Nonsmooth Radial Scattering Solutions of a Loglog Energy-Supercritical Schrödinger Equation

2018 ◽  
Vol 2020 (8) ◽  
pp. 2501-2541
Author(s):  
Tristan Roy
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
A Priori ◽  
Time Interval ◽  
Type Estimate ◽  
A Priori Bound ◽  
Long Time ◽  
Supercritical Nonlinearity ◽  
Gamma S ◽  
Radial Scattering

Abstract We prove scattering of solutions of the loglog energy-supercritical Schrödinger equation $i \partial _{t} u + \triangle u = |u|^{\frac{4}{n-2}} u g(|u|)$ with $g(|u|) := \log ^{\gamma } {( \log{(10+|u|^{2})} )}$, $0 &lt; \gamma &lt; \gamma _{n}$, n ∈ {3, 4, 5}, and with radial data $u(0) := u_{0} \in \tilde{H}^{k}:= \dot{H}^{k} (\mathbb{R}^{n})\,\cap\,\dot{H}^{1} (\mathbb{R}^{n})$, where $\frac{n}{2} \geq k&gt; 1 \left(\text{resp.}\,\frac{4}{3}&gt; k &gt; 1\right)$ if n ∈ {3, 4} (resp. n = 5). The proof uses concentration techniques (see e.g., [ 2, 12]) to prove a long-time Strichartz-type estimate on an arbitrarily long time interval J depending on an a priori bound of some norms of the solution, combined with an induction on time of the Strichartz estimates in order to bound these norms a posteriori (see e.g., [ 8, 10]). We also revisit the scattering theory of solutions with radial data in $\tilde{H}^{k}$, $k&gt; \frac{n}{2}$, and n ∈ {3, 4}; more precisely, we prove scattering for a larger range of $\gamma$ s than in [ 10]. In order to control the barely supercritical nonlinearity for nonsmooth solutions, that is, solutions with data in $\tilde{H}^{k}$, $k \leq \frac{n}{2}$, we prove some Jensen-type inequalities.

scholarly journals Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chao Yang
Keyword(s):  
Global Existence ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Term ◽  
Invariant Sets ◽  
Well Posedness ◽  
Sharp Condition ◽  
The Cauchy Problem ◽  
Inhomogeneous Nonlinear Schrödinger Equation ◽  
Time Blowup

<p style='text-indent:20px;'>This paper studies the Cauchy problem of Schrödinger equation with inhomogeneous nonlinear term <inline-formula><tex-math id="M1">\begin{document}$ V(x)|\varphi|^{p-1}\varphi $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. For the case <inline-formula><tex-math id="M3">\begin{document}$ p&gt;1+\frac{4(1+\varepsilon_0)}{n} (0&lt;\varepsilon_0&lt;\frac{2}{n-2}) $\end{document}</tex-math></inline-formula>, by introducing a potential well, we obtain some invariant sets of solution and give a sharp condition of global existence and finite time blowup of solution; for the case <inline-formula><tex-math id="M4">\begin{document}$ p&lt;1+\frac{4}{n} $\end{document}</tex-math></inline-formula>, we obtain the global existence of solution for any initial data in <inline-formula><tex-math id="M5">\begin{document}$ H^1 (\mathbb{R}^n) $\end{document}</tex-math></inline-formula>.</p>

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