THE BLOW-UP PROBLEM FOR A CLASS OF QUASILINEAR SCHRÖDINGER EQUATIONS

Qi Guo; Boling Guo

doi:10.11948/20250215

2026 Volume 16 Issue 5

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Article navigation > Journal of Applied Analysis & Computation > 2026 > 16(5): 2705-2719

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Qi Guo, Boling Guo. THE BLOW-UP PROBLEM FOR A CLASS OF QUASILINEAR SCHRÖDINGER EQUATIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2705-2719. doi: 10.11948/20250215

Citation:

Qi Guo, Boling Guo. THE BLOW-UP PROBLEM FOR A CLASS OF QUASILINEAR SCHRÖDINGER EQUATIONS[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2705-2719. doi: 10.11948/20250215

THE BLOW-UP PROBLEM FOR A CLASS OF QUASILINEAR SCHRÖDINGER EQUATIONS

Qi Guo^1, ,,
Boling Guo^2,

1.
School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China
2.
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Author Bio: Email: gbl@iapcm.ac.cn(B. Guo)
Corresponding author: Email: guoqi23@emails.bjut.edu.cn(Q. Guo)

Abstract

Abstract

We consider the blow-up results of the solution in $ W^{2, 2}(\mathbb{R}^N) $ for the following quasilinear Schrödinger equation
$ \begin{align*} \begin{cases} { \begin{array}{ll} iu_t+\Delta u+2uh'(|u|^2)\Delta h(|u|^2)+uf(|u|^2)=0, \qquad x\in \mathbb{R}^N, \\ u(x, 0)=u_0(x), \qquad x\in \mathbb{R}^N, \end{array} } \end{cases} \end{align*} $
where $ h $ and $ f $ are real functions which related to various physical models. We prove that the $ W^{2, 2}(\mathbb{R}^N) $ solutions must blow up if $ |x|u_0\in L^2(\mathbb{R}^N) $(finite variance), and we give the upper bound of the blow-up time. We also show that without the finite variance assumption, the radial symmetric solutions in $ W^{2, 2}(\mathbb{R}^N) $ must blow up in finite time for the whole class of initial data with strictly negative energy.
- Quasilinear Schrödinger equation /
- blow up /
- radial symmetric solution
MSC: 35Q55, 35B44, 35B06

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References

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