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Modified scattering for higher-order nonlinear Hartree-type equations

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Abstract

We study the Cauchy problem for the higher-order nonlinear 3D Hartree-type equation

$$\begin{aligned} \left\{ \begin{array} [c]{ll} i\partial _{t}u+\frac{1}{2}\Delta u-\frac{1}{4}\Delta ^{2}u=\left( \left| x\right| ^{-1}*\left| u\right| ^{2}\right) u,&{}{}\quad t>0, x\in {\mathbb {R}}^{3},\\ u\left( 0,x\right) =u_{0}\left( x\right) ,&{}{}\quad x\in {\mathbb {R}}^{3}, \end{array} \right. \end{aligned}$$

where \(*\) denotes the convolution in \({\mathbb {R}}^{3}\). Cubic Hartree-type nonlinearity usually behaves critically for large time in three-space-dimensional case. In the present paper, we develop the factorization technique for the case of the higher-order Hartree-type equation (1.1) to show that the large time asymptotics of solutions to the Cauchy problem for the higher-order nonlinear Hartree-type equation has a modified character.

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Acknowledgements

We are grateful to unknown referees for many useful suggestions and comments. The work of P.I.N. is partially supported by CONACYT Project 283698 and PAPIIT Project IN103221.

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Authors and Affiliations

  1. Instituto Tecnologico de Morelia, Avenida Tecnologico #1500, Lomas de Santiaguito, 58120, Morelia, Michoacán, Mexico

    Beatriz Juarez-Campos & Hector F. Ruiz-Paredes

  2. Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), 58089, Morelia, Michoacán, Mexico

    Pavel I. Naumkin

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  1. Beatriz Juarez-Campos
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  2. Pavel I. Naumkin
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  3. Hector F. Ruiz-Paredes
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Correspondence to Pavel I. Naumkin.

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Juarez-Campos, B., Naumkin, P.I. & Ruiz-Paredes, H.F. Modified scattering for higher-order nonlinear Hartree-type equations. Z. Angew. Math. Phys. 72, 92 (2021). https://doi.org/10.1007/s00033-021-01529-3

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  • DOI: https://doi.org/10.1007/s00033-021-01529-3

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