scholarly journals Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xuan Liu ◽  
Ting Zhang
Keyword(s):  
Schrödinger Equation ◽  
Fourth Order ◽  
Finite Time ◽  
Schrodinger Equation ◽  
Well Posedness ◽  
Complex Coefficient ◽  
Finite Time Blowup ◽  
Time Blowup

scholarly journals Finite-time blowup for a Schrödinger equation with nonlinear source term

2019 ◽  
Vol 39 (2) ◽  
pp. 1171-1183 ◽  
Author(s):  
Thierry Cazenave ◽  
◽  
Yvan Martel ◽  
Lifeng Zhao ◽  
◽  
...  
Keyword(s):  
Schrödinger Equation ◽  
Finite Time ◽  
Schrodinger Equation ◽  
Source Term ◽  
Nonlinear Source ◽  
Finite Time Blowup ◽  
Time Blowup

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>

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].

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