# American Institute of Mathematical Sciences

May  2014, 34(5): 2261-2281. doi: 10.3934/dcds.2014.34.2261

## Scattering theory for the wave equation of a Hartree type in three space dimensions

 1 Graduate School of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan

Received  December 2012 Revised  August 2013 Published  October 2013

The paper concerns a scattering problem of the wave equation of a Hartree type with small initial data with fast decay. The equation is $\partial_t^2 u - \Delta u = V_1(x)u+ (V_2\ast |u|^{p-1})u , \qquad t\in {\bf R}, \; x \in {\bf R}^3,$ where $p\ge 3, \; V_1(x)=O(|x|^{-\gamma_1})$ with $\gamma_1>0$ as $|x|\to\infty, \; V_2(x) = \pm |x|^{-\gamma_2}$ with $\gamma_2>0$. We prove the existence of scattering operators under almost optimal conditions on the potentials and initial data in terms of decay, using pointwise estimates. Our result generalizes the one by [14, 15] for the case $p=3$.
Citation: Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261
##### References:
 [1] F. Asakura, Existence of a global solution to a semi-linear wave equation with slowly decreasing initial data in three space dimensions, Comm. Partial Differential Equations, 11 (1986), 1459-1487. doi: 10.1080/03605308608820470. [2] K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcialaj Ekvacioj, 43 (2000), 559-588. [3] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. doi: 10.1007/BF01647974. [4] P. Karageorgis and K. Tsutaya, On the asymptotic behavior of nonlinear waves in the presence of a short-range potential, Manuscripta Math., 119 (2006), 323-345. doi: 10.1007/s00229-005-0620-z. [5] P. Karageorgis and K. Tsutaya, On the Asymptotic Behavior of Solutions of the Wave Equation of Hartree Type,, in preparation., (). [6] P. Karageorgis and K. Tsutaya, Existence and Blow Up for A Hartree-Type Wave Equation,, in preparation., (). [7] H. Kubo, On Point-Wise Decay Estimates for the Wave Equation and Their Applications, Dispersive nonlinear problems in mathematical physics, 123-148, Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta, 2004. [8] G. Perla Menzala and W. A. Strauss, On a wave equation with a cubic convolution, J. Diff. Eq., 43 (1982), 93-105. doi: 10.1016/0022-0396(82)90076-6. [9] K. Mochizuki and T. Motai, On Small Data Scattering for Some Nonlinear Wave Equations, Patterns and waves, 543-560, Stud. Math. Appl., 18, North-Holland, Amsterdam, 1986. doi: 10.1016/S0168-2024(08)70145-0. [10] K. Mochizuki, On small data scattering with cubic convolution nonlinearity, J. Math. Soc. Japan, 41 (1989), 143-160. doi: 10.2969/jmsj/04110143. [11] H. Pecher, Scattering for semilinear wave equations with small data in three space dimensions, Math. Z., 198 (1988), 277-289. doi: 10.1007/BF01163296. [12] W. A. Strauss, Nonlinear invariant wave equations, Lecture Notes in Phys., 73, Springer, Berlin-New York, (1978), 197-249. [13] W. A. Strauss and K. Tsutaya, Existence and blow up of small amplitude nonlinear waves with a negative potential, Discrete Continuous Dynam. Systems, 3 (1997), 175-188. doi: 10.3934/dcds.1997.3.175. [14] K. Tsutaya, Global existence and blow up for a wave equation with a potential and a cubic convolution, Nonlinear Analysis and Applications: to V. Lakshmikantham on his 80th Birthday. Vol. 1, 2, 913-937, Kluwer Acad. Publ., Dordrecht, 2003. [15] K. Tsutaya, Scattering theory for a wave equation of Hartree type, Differential & Difference Equations And Applications, 1061-1065, Hindawi Publ. Corp., New York, 2006. [16] K. Tsutaya, Weighted estimates for a convolution appearing in the wave equation of Hartree type,, to appear in J. Math. Anal. Appl.., ().

show all references

##### References:
 [1] F. Asakura, Existence of a global solution to a semi-linear wave equation with slowly decreasing initial data in three space dimensions, Comm. Partial Differential Equations, 11 (1986), 1459-1487. doi: 10.1080/03605308608820470. [2] K. Hidano, Small data scattering and blow-up for a wave equation with a cubic convolution, Funkcialaj Ekvacioj, 43 (2000), 559-588. [3] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268. doi: 10.1007/BF01647974. [4] P. Karageorgis and K. Tsutaya, On the asymptotic behavior of nonlinear waves in the presence of a short-range potential, Manuscripta Math., 119 (2006), 323-345. doi: 10.1007/s00229-005-0620-z. [5] P. Karageorgis and K. Tsutaya, On the Asymptotic Behavior of Solutions of the Wave Equation of Hartree Type,, in preparation., (). [6] P. Karageorgis and K. Tsutaya, Existence and Blow Up for A Hartree-Type Wave Equation,, in preparation., (). [7] H. Kubo, On Point-Wise Decay Estimates for the Wave Equation and Their Applications, Dispersive nonlinear problems in mathematical physics, 123-148, Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta, 2004. [8] G. Perla Menzala and W. A. Strauss, On a wave equation with a cubic convolution, J. Diff. Eq., 43 (1982), 93-105. doi: 10.1016/0022-0396(82)90076-6. [9] K. Mochizuki and T. Motai, On Small Data Scattering for Some Nonlinear Wave Equations, Patterns and waves, 543-560, Stud. Math. Appl., 18, North-Holland, Amsterdam, 1986. doi: 10.1016/S0168-2024(08)70145-0. [10] K. Mochizuki, On small data scattering with cubic convolution nonlinearity, J. Math. Soc. Japan, 41 (1989), 143-160. doi: 10.2969/jmsj/04110143. [11] H. Pecher, Scattering for semilinear wave equations with small data in three space dimensions, Math. Z., 198 (1988), 277-289. doi: 10.1007/BF01163296. [12] W. A. Strauss, Nonlinear invariant wave equations, Lecture Notes in Phys., 73, Springer, Berlin-New York, (1978), 197-249. [13] W. A. Strauss and K. Tsutaya, Existence and blow up of small amplitude nonlinear waves with a negative potential, Discrete Continuous Dynam. Systems, 3 (1997), 175-188. doi: 10.3934/dcds.1997.3.175. [14] K. Tsutaya, Global existence and blow up for a wave equation with a potential and a cubic convolution, Nonlinear Analysis and Applications: to V. Lakshmikantham on his 80th Birthday. Vol. 1, 2, 913-937, Kluwer Acad. Publ., Dordrecht, 2003. [15] K. Tsutaya, Scattering theory for a wave equation of Hartree type, Differential & Difference Equations And Applications, 1061-1065, Hindawi Publ. Corp., New York, 2006. [16] K. Tsutaya, Weighted estimates for a convolution appearing in the wave equation of Hartree type,, to appear in J. Math. Anal. Appl.., ().
 [1] Tarek Saanouni. Energy scattering for the focusing fractional generalized Hartree equation. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3637-3654. doi: 10.3934/cpaa.2021124 [2] Changhun Yang. Scattering results for Dirac Hartree-type equations with small initial data. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1711-1734. doi: 10.3934/cpaa.2019081 [3] Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431 [4] Kenji Nakanishi. Modified wave operators for the Hartree equation with data, image and convergence in the same space. Communications on Pure and Applied Analysis, 2002, 1 (2) : 237-252. doi: 10.3934/cpaa.2002.1.237 [5] Qingxuan Wang, Binhua Feng, Yuan Li, Qihong Shi. On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1225-1247. doi: 10.3934/cpaa.2022017 [6] Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066 [7] Xiaomei Chen, Xiaohui Yu. Liouville type theorem for Hartree-Fock Equation on half space. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2079-2100. doi: 10.3934/cpaa.2022050 [8] Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557 [9] Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems and Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004 [10] Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048 [11] Jeong Ja Bae, Mitsuhiro Nakao. Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 731-743. doi: 10.3934/dcds.2004.11.731 [12] Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134 [13] Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. On small data scattering of Hartree equations with short-range interaction. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1809-1823. doi: 10.3934/cpaa.2016016 [14] Anudeep Kumar Arora. Scattering of radial data in the focusing NLS and generalized Hartree equations. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6643-6668. doi: 10.3934/dcds.2019289 [15] Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903 [16] Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems and Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013 [17] John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems and Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181 [18] Gabriella Pinzari. Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result. Electronic Research Announcements, 2015, 22: 55-75. doi: 10.3934/era.2015.22.55 [19] Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems and Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121 [20] Xiaoli Feng, Meixia Zhao, Peijun Li, Xu Wang. An inverse source problem for the stochastic wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 397-415. doi: 10.3934/ipi.2021055

2020 Impact Factor: 1.392