带部分调和势的非齐次非线性 Schrödinger 方程的爆破解

Abstract

Abstract:

This paper is devoted to the Cauchy problem of inhomogeneous nonlinear Schrödinger equation in the presence of a partial confinement, which is an important model in Bose-Einstein condensates. Combining the variational characterization of the ground state of a nonlinear elliptic equation and the conservations of mass and energy, we first obtain a global solution and show the existence of blow-up solutions for some special initial data by scaling techniques. Then, we study the $L^2$-concentration phenomenon for the blow-up solutions. Finally, we apply the variational arguments connected to the above ground state to investigate the dynamics of $L^2$-minimal blow-up solutions, i.e., the limiting profile, mass-concentration and blow-up rate of the blow-up solutions with minimal mass. We extend the global existence and blow-up results of Zhang[34] to the case of inhomogeneous nonlinearities and improve partial results of Pan and Zhang[23] to space dimensions $N\geq2$ in the inhomogeneous case.

Key words: Inhomogeneous nonlinear Schr?dinger equation, Partial confinement, Blow-up, Mass-concentration, Limiting

CLC Number:

O175.23

Jian Hui, Gong Min, Wang Li. On the Blow-Up Solutions of Inhomogeneous Nonlinear Schrödinger Equation with a Partial Confinement[J].Acta mathematica scientia,Series A, 2023, 43(5): 1350-1372.

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References

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On the Blow-Up Solutions of Inhomogeneous Nonlinear Schrödinger Equation with a Partial Confinement

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